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FIVE-FIG-UBE TABLES

OF

MATHEMATICAL FUNCTIONS

COMPRISING

TABLES OP LOGARITHMS, POWERS OF NUMBERS,

TRIGONOMETRIC, ELLIPTIC, AND OTHER

TRANSCENDENTAL FUNCTIONS

BY

JOHN BORTHWIOK DALE, M.A.

A88I8TANT PROFESSOR OF MATHEMATICS IN KTNOlB COLLBGft,
UNIVERSITY OF LONDON

EIGHTH IMPRESSION

LONDON

EDWARD ARNOLD
41 AND 43 MADDOX STREET, BOND STREET, W.

PREFACE.

THIS collection of Tables has been formed chiefly with a view
to meet the requirements of workers in Physical Science and
Applied Mathematics. It includes many tables which have
not hitherto found a place in such collections, while those
tables which are of use only in Navigation have been excluded.

For ordinary laboratory work four-figure tables of logarithms
are at present generally employed. The small space they
occupy renders them convenient to use, but they suffer from
the defect that the accumulation of small errors frequently
makes even the third significant figure in the result unreliable.
For this reason five-figure tables are preferable.

Four-figure tables give directly the logarithm for three
significant figures, and, by differences, the logarithm for four;
while five-figure tables usually give directly the logarithm for
four significant figures, and so occupy ten times the space of
a four-figure table. A glance, however, at the table in the
following pages will show that it is only in the first third
of it that the rapid variation of the differences renders a direct
tabulation for four significant figures desirable. This has not
been done, but merely the mean differences for each row have
been tabulated, and these differences may be used when results
accurate to four figures only are desired. Results accurate to
five figures are to be obtained by using the Table of Anti-
l^garithms; for it is precisely in that part of the Table of

iv PREFACE.

Logarithms where the variation is most rapid that the variation
in the Table of the Inverse Function is slowest, and vice versa.

The conjoint use of these Tables, therefore, offers all the
accuracy obtainable with the usual five-figure tables, with the
expenditure of little more labour than is involved in consulting
a four-figure table.

The principle stated above applies to several other pairs of
Tables, such as those of Squares, Square Boots, Cubes, and Cube
Eoots ; and by taking advantage of it, it has become possible
to include a great number of tables within a moderate compass,
without sacrificing convenience or accuracy.

Several of the Tables have been constructed specially for
this collection, and some others appear for the first time in
their present form. Every care has been taken, and it is
hoped they will be found free from error.

J. BOETHWICK DALK

NEW MALDEN, SURREY,
December, 1903.

CONTENTS.

I. COMMON LOGARITHMS 1

II. ANTILOGAHITHMS 4

III. SQUARES * * 7

IV. SQUARE BOOTS . 10

V. CUBES .,.....*.. 16

VL CUBE BOOTS 19

VII. BECIPROCALS , ,28

VIII. NATURAL LOGARITHMS OP 10 n AND 10"" . . .31

IX. RECIPROCALS OF FACTORIALS ...... 31

X. BERNOULLI'S NUMBERS .31

XI. NATURAL SINES AND COSINES 32

XII. NATURAL TANGENTS AND COTANGENTS . . . . 36

XIII. NATURAL SECANTS AND COSECANTS 40

XIV. LOGARITHMIC SINES AND COSINES ..... 44
XV. LOGARITHMIC TANGENTS AND COTANGENTS ... 48

XVI. LOGARITHMIC SECANTS AND COSECANTS . . . .52

XVII. CIRCULAR, OB BADIAN MEASURE 56

XVIII. CONVERSION OF TIME AND ANGULAR MEASURB . .59
XIX. DECIMAL EQUIVALENTS ....... 60

XX. NATURAL, OR NAPERIAN LOGARITHMS Gl

XXL EXPONENTIAL FUNCTIONS, HYPERBOLIC SINES AND COSINES . 64

XXII. Log u e- 66

XXIII. (Log, tan (45 + =^- 3 ^ 67

vi CONTENTS.

PAQB

XXIV. ELLIPTIC FUNCTIONS. F (9, <j>) 68

XXV ELLIPTIC FUNCTIONS. E (0, <j>) 72

XXVI. COMPLETE ELLIPTIC FUNCTIONS ...... 76

XXVII. GAMMA FUNCTION . 77

XXVIII. LOGARITHMS OF FACTORIALS 77

XXIX. ZONAL SURFACE HARMONICS 78

XXX. BESSEL FUNCTIONS. J (a?) AND J^a) 80

XXXI. BESSEL FUNCTIONS. J N (a) c 81

XXXII. BESSEL FUNCTIONS. !() 82

XXXIII. BESSEL FUNCTIONS. ROOTS OF J n (a;), J (), AND J,() . 83

XXXIV. BESSEL FUNCTIONS. VALUES OF J, + 1(&) . . . .83

XXXV. NORMAL ERROR OR PROBABILITY INTEGRAL . '84
XXXVI. EXPONENTIAL, COSINE AND SINE INTEGRALS . . . .85

XXXVII. BINOMIAL COEFFICIENTS FOR INTERPOLATION BY DIFFERENCES 87

XXXVIII. PROPORTIONAL PARTS 88

XXXIX. NUMBERS USED IN CALCULATIONS 90

EULER'S NUMBERS. E w 91

EULER'S CONSTANT 91

VALUES OF S n , W| T n< V w 92

ENGLISH AND METRIC EQUIVALENTS . . . 92

EIVE-FIGUBE TABLES

OP

MATHEMATICAL FUNCTIONS.

EXPLANATION OF THE TABLES.

I. LOGARITHMS OF NUMBERS.

THE logarithms here tabulated are logarithms to the base 10,
known as Briggian or Common Logarithms.

The logarithm of a number consists of two parts, an integral
part and a fractional part. The integral part is called the
index, or characteristic ; the fractional part is called the mantissa.

The characteristic of a number greater than 1 is the number
one less than the number of digits in the integral part of the
given number.

The characteristic of a number less than 1 is negative and
is the number one greater than the number of zeros following
the decimal point.

Ex. The characteristics of 5387, 5387, 0-0005387, are 3, 1, and -4
respectively.

The mantissa of all numbers having the same significant
figures are the same.

To find the mantissa of a given number.

(a) If the number contains three significant figures, find
the first two figures in the left-hand column of the table, and
the third figure in the top row. The mantissa required will

viii FIVE-FIGURE TABLES OF

be found in the row containing the first two figures and in the
column under the third figure.

Ex. Log 53-8 = 1-73078. Log 0-00538 = 3-73078.

The mantissa is always positive, and when it is preceded by
a negative characteristic, the negative sign is written above the
characteristic.

(6) If the number contains four or five significant figures,
find the mantissa corresponding to the first three figures as
before. In the top row under Mean Differences find the fourth
figure, and in the column under it, in the row containing the
first two figures will be found the difference to be added to
the mantissa. If there is a fifth figure, take the nearest tenth
of the number corresponding to it in the Mean Difference
columns, and add to the mantissa.

Ex. To find log 53-874.

Log 53-8 = 1-73078

Diff. for 7 = 57

Diff.for4 = 3

Log 53-874 = 1-73138

Near the beginning of the table the differences are large
and change rapidly, and unless the third significant figure is
4 or 5, the fifth figure of the mantissa obtained by using the
mean differences will be incorrect. In this case the Table ot
Antilogarithms may be used, or the Principle of Proportional
Parts.

Ex. To find log 120-73.

Log 120 = 2-07918
Log 121 = 2-08279

Diff. for 1 = 361

Diff. for 0-73 = 361 x 0-73 = 264

Log 120-73 =2-08182

The product 361 x 0*73 may be readily found by the
Table of Proportional Parts at the end of the book.

To find the number corresponding to a given logarithm.

If the mantissa is found exactly in the tables, the first two
significant figures of the number will be found in the same
row at the left hand, and the third figure at the top of the

MATHEMATICAL INUNCTIONS. ix

column. The decimal point may then be placed by noting the
value of the characteristic.

If the mantissa is not found exactly in the tables, take out
the next lower, and the three figures of the number belonging
to it. Find the difference between this logarithm and the
given one, and find the number corresponding to this difference
in the same row under Mean Differences. If the difference is
remainder by 10, and find the number corresponding to the
product.

Ex. Given logarithm = 1-73138

Next lower logarithm = 1-73078 = log 53-8

Diff. = CO

Next lower difference = 57 = diff. for 7

3

30 = diff. for 4
Therefore number is 53'874

II. ANTILOGAKITHMS.

In this table will be found the significant figures corre-
sponding to given logarithms. The first two figures of the
mantissa are found in the left-hand column, and the third in
the top row. If there are more than three significant figures
in the mantissa the corresponding differences will be found in
the columns of Mean Differences.

Ex. Given.logarithm = 2*08182

2-081 = log 120-50

Diff. for 8 = 22

Diff. for 2 = 1

2-08182 = log 120-73

Near the end of the table, the differences are large and
change rapidly. Hence, unless the third figure of the mantissa
be 4 or 5, use the Table of Logarithms or the Principle of
Proportional Parts.

Use of the Tables of Logarithms and Antilogarithms.

These tables may be used to perform multiplication, division,

x FIVE-FIGURE TABLES OF

involution, and root extraction in accordance with the following
propositions :

(1) Log A + log B = log (A x B)
(ii.) Log A - log B = log (A ~- E)
(iii.) n log A = log A n

(iv.) i log -4 = log <X^4

71

NOTE.- When multiplying or dividing a logarithm whoso
characteristic is negative, it is necessary to bear in mind that
the mantissa is positive.

Ex. (i.) 3-73138 x 3 = (-3 + 0-73138) x 3 = -9 + 2-19414 = 7-19414
Ex. (ii

III., IV., V., VI., VII. POWERS OF NUMBERS.

These tables give the reciprocals, squares, and cubes of
numbers between 1 and 10; the square roots of numbers
between 1 and 100 ; and the cube roots of numbers between
1 and 1000.

The method of using them is essentially the same as that
of using the Tables of Logarithms and Antilogarithms.

The cubes of numbers containing four or five significant
figures should be found by logarithms or by the Tables of Cube
Boots.

XL, XII., XIII. TKIGONOMETRICAL FUNCTIONS.

These tables give the values of the natural sines, tangents,
etc., for every third minute of the quadrant. The difference
columns give the corrections for 1' and 2',

The left-hand column of degrees and the top row of minutes
are to be followed when sines, tangents, and secants are taken
out

The right-hand column of degrees and bottom row of
minutes are to be used when cosines, cotangents, and cosecants
are taken out

MATHEMATICAL FUNCTIONS. xi

Note that for the three co-functions, differences are to be
subtracted.

The integral part and decimal point are printed only in
the columns headed 0' and 33'.

XIV., XV., XVI. LOGARITHMIC TRIGONOMETRICAL FUNCTIONS.

These tables give the logarithms of the numbers contained
in the three preceding tables increased by 10, in order to avoid
negative characteristics.

The characteristic and decimal point are printed only in the

Ex. Find sin 37 34' 12" and log sin 37 34' 12".

Sin 37 33' = 0-60945 Log sin 37 33' = 9-78494

Diff.forl' = 23 Diff. for 1' = 16

Diff. for 12'' = 5 Diff. for 12" = 3

Sin 37 34' 12" = 0-60973 Log sin 37 34' 12" = 9-78513

When the values of the tangent, cotangent, secant, and
cosecant are large and change rapidly, and are not given exactly
in the tables, they should be found from the sine and cosine.

Small Angles.

The differences for the logarithmic sines and tangents of
small angles are large and change rapidly. The values of the
logarithms can be found by the following rules :

(1) To obtain the log sine.

Eeduce the angle to seconds, and to the logarithm of the
number so obtained add 4*68557. From the sum subtract J of
the log secant diminished by 10.

(2) To obtain the log tangent.

Eeduce the angle to seconds, and to the logarithm of the
number so obtained add 4'68557. To the sum add of the log
secant diminished by 10.

(3) To obtain the angle, given the log sine.

To the log sine add 5*31442 and J of the corresponding log
secant diminished by 10. The sum is the logarithm of the
number of seconds in the angle.

xii FIVE-FIGURE TABLES OF

(4) To obtain the angle, given the log tangent.

To the log tangent add 5-31442 and subtract | of the corre-
sponding log secant diminished by 10. The remainder is the
logarithm of the number of seconds in the angle.

VIII., XX. NATURAL LOGARITHMS.

This table gives the logarithms of numbers to the base
e (= 27182818 . . .).

Every number may be expressed as the product of a
number between 1 and 10, and some power of 10. The
logarithm of the given number is the sum of the logarithms
of the two factors.

Ex. To find log, 735-2 and log, 0-0007352.

735-2 = 7-352 x 10* 0-0007352 = 7-352 x 10'*

Log, 7-35 = 1-99470 Log, 7-35 = 1-99470

Diff.for2 = 27 Diff. for 2 27

Log, 10 2 = 4-60517 Log, 10-* = 10-78966

Log, 735-2 = 6-60014 Log 0-0007352 = 8-78463

XXI., XXII., XXIII. EXPONENTIAL AND HYPERBOLIC
FUNCTIONS.

The values of e x , e~ x t cosh x, sinh x are tabulated at intervals
of 0*01 from x = 1 to x = 3, and at intervals of 0*1 from x = 3
to x = 6.

The values of e x and e~* for greater values of x may be
found from the table for Iog 10 e*. When e* = y y x = log. y.
Hence these tables may be regarded as tables of antilogarithms
for the base e.

The hyperbolic functions are defined by the relations

cosh x = (e* + "*) s i n h x = l( e * e ~*)

tanh x = sinh x -7- cosh x, coth x = 1 -f- tanh x,

sech x = 1 -7- cosh x, cech x = 1 -f- sinh x.

When x is greater than 6, it is sufficiently accurate to take
cosh x = sinh x = \e*. The hyperbolic functions m^y be

MATHEMATICAL FUNCTIONS. xiii

connected with the trigonometric functions by means of the table
for u as lo tan

cosh w = sec x, sinh w = tan x> tanh u = sin x.

XXIV., XXV., XXVI. ELLIPTIC FUNCTIONS.

<*>

f cfo?

These tables give the values of F(k, 6) = I

I

J

. .

v 1 - k* sin 2 a?

_

and EQc, 0) = J y^i _ / C 2 S j n 2 ^ ^ f or every degree of the

^ o

amplitude ^>, and every fifth degree of 0, when sin =* k, the
moefa^s.

Table XXI. gives the values of the complete elliptic functions

F l = F (k, |) and E^- E (k, ^), for every degree of

9 = sirr 1 ^.

(N.B. j^j is now generally denoted by K.)
If u = F(k, 0), \$ = am(^, A).

k) = sin ^>, cn(w, k) = cos ^>, dn(w,*)

Note sn(^, 1) = tanh u } cn(u, 1) = dn(?^, 1) = sech u.

XXIX. ZONAL SURFACE HARMONICS.

This table gives the values of the first seven harmonics for
values of the argument x at intervals of O'Ol.

P Q (x) = 1. Pi(*) = x.

- 1). PB() = i(5^ - 3a?).

3).

- 5).
P 7 (a?) = ^,

xiv FIVE-FIGURE TABLES OF

XXX. XXXIV. BESSEL FUNCTIONS.

These tables give the values of J n (x) and also of
I n (x) as i" n j n (tx) where i = \/ 1.

XXXV. NORMAL ERROR OR PROBABILITY INTEGRAL.

2 r*

This table gives the values of the integral / = /- I e~ t2 dt.

v 71 " J

XXVII. GAMMA FUNCTIONS.
This table gives the values of Iogi r(#) = log ]

for values of x between 1 and 2.

When x is greater than 2, the value may be obtained by
using the formula of reduction

r(x + 1) = ar(a)

When x is large, the value of log, r(x + 1) is approximately
given by

log. r(x + 1) = log. \X2ir + (a + 4) lo & ~ a +

1.2.a; 3 . 4.a? 5 . 6 .

XXVIII. LOGARITHMS OF FACTORIALS.

This table gives the values of log II(?i) = log x\ for integral
values of x from x = 1 to x = 100,
When x is integral Y(x + 1) = xl
When x is large

MATHEMATICAL FUNCTIONS. xv

XXXVI. EXPONENTIAL, COSINE, AND SINE INTEGRALS.
These tables give the values of the exponential integrals

X "X

J\$i(x) = I du, and JEi(-x) = I du\ of the cosine integral

* -co J -

X X

f cos u , , - , . . f sin M ,
i 0^ an d O f the sine integral I du.

1 lv I Iv

J /

z

The logarithmic integral Li(x) = I j = JSi(log x).

J o

Hence its values may be obtained from those of the exponential
integral.

X. BERNOULLI'S NUMBERS.
These numbers are the coefficients in the expansion of

x e* + 1 . ,1 r

m the torm
2 e* - 1

[N.B. The notation varies, the series of numbers being
sometimes written B\ 9 B 3 , B\$ . . . -#a-i is the same as B n in
the table here printed.]

XXXVII. BINOMIAL COEFFICIENTS FOR INTERPOLATION BY

DIFFERENCES.

This table gives the values of the coefficients -^ ~- and

X .

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1 . -& . o

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where AI/() =/(^ + 1) - /(*)

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SINES.

MKAJN

DlKF.

83'

36'

89'

42'

45'

48'

51'

54'

57'

60'

1'

2'

00960

01047

01134

01222

01309

01396

01483

01571

01658

01745

89

29

58

02705

02792

02879

02967

03054

03141

03228

03316

03403

03490

88

29

58

04449

04536

04623

04711

04798

04885

04972

05059

05146

05234

87

29

58

06192

06279

06366

06453

06540

06627

06714

06802

06889

06976

86

29

58

07933

08020

08107

08194

08281

08368

08455

08542

08629

08716

85

29

68

09671

09758

09845

09932

10019

10106

10192

10279

10366

10453

84

29

58

11407

11494

11580

11667

11754

11840

11927

12014

12100

12187

83

29

58

13139

13226

13312

13399

13485

13572

13658

13744

13831

13917

82

29

58

14867

14954

15040

15126

15212

15299

15385

16471

15557

16643

81

29

58

16591

16677

16763

16849

16935

17021

17107

17193

17279

17365

80

29

57

18309

18395

18481

18567

18652

18738

18824

18910

18995

19081

79

29

57

20022

20108

20193

20279

20364

20450

20535

20620

20706

20791

78

29

57

21729

21814

21899

21985

22070

22155

22240

22325

22410

22495

77

28

57

23429

23514

23599

28684

237G9

23853

23938

24023

24108

24192

76

28

57

25122

25207

25291

25376

25460

25545

25629

25713

25798

25882

75

28

56

26808

26892

26976

27060

27144

27228

27342

27396

27480

27564

74

28

56

28485

28569

28652

28736

28820

28903

28987

29070

29154

29237

73

28

56

30154

30237

30320

30403

30486

30570

30653

30736

30819

30902

72

28

55

31813

31896

31979

32061

32144

32227

32309

32392

32474

32557

71

28

55

33463

33545

33G27

33710

33792

33874

33956

34038

34120

34202

70

27

55

35102

35184

35266

35347

35429

35511

35592

35674

35755

35837

69

27

54

36731

36812

36894

36975

37056

37137

37218

37299

37380

37461

68

27

54

38349

38430

38510

38591

38671

38752

38832

38912

38993

39073

67

27

64

39955

40035

40115

40195

40275

40355

40434

40514

40594

40674

66

27

63

41549

41628

41707

41787

41866

41945

42024

42104

42183

42262

65

26

53

43130

43209

43287

43366

43445

43523

43602

43680

43759

43837

64

26

52

44698

44776

44854

44932

45010

45088

45166

45243

45321

45399

63

26

62

46252

46330

46407

46484

46561

46639

46716

46793

46870

46947

62

26

51

47793

47869

47946

48022

48099

48175

48252

48328

48405

48481

61

26

51

49318

49394

49470

4954G

49622

49697

49773

49849

49924

50000

60

25

51

50829

50904

50979

51054

51129

51204

51279

51354

61429

51504

59

25

50

52324

52399

52473

52547

52621

52696

52770

52844

52918

52992

58

25

50

53804

53877

53951

54024

54097

54171

54244

54317

54391

54464

57

24

49

55266

55339

55412

55484

55557

55630

55702

55775

55847

55919

56

24

48

56713

56784

56856

56928

57000

57071

57143

57215

57286

67358

55

24

48

58141

58212

58283

58354

58425

58496

58567

58637

58708

58779

54

24

47

59552

59622

69693

69763

59832

51)902

59972

60042

60112

60182

53

23

47

60945

61015

61084

61153

61222

61291

61360

61429

61497

61566

52

23

46

62320

62388

62456

62524

62592

626GO

62728

62796

62864

62932

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23

45

63675

^3742

63810

63877

63944

64011

64078

64145

64212

64279

50

22

45

65011

65077

65144

65210

65276

65342

65408

65474

65540

65C06

49

22

44

66327

66393

66458

66523

66588

66653

66718

66783

66848

66913

48

22

43

67623

.67688

67752

67816

67880

67944

68008

68072

68136

68200

47

21

43

68899

68962

69025

69088

69151

69214

69277

69340

69403

69466

46

21

42

70153

70215

70277

70339

70401

70463

70525

70587

70649

70711

45

21

41

27'

24'

21'

18'

15'

12'

&

ff

y

0'

r

2'

MEAN

COSINES. Drrp,

34

NATURAL

MEAN

DIFF.

<y

3'

6*

0'

12'

15'

18'

21'

24*

27'

30'

r

2'

45

70711

70772

70834

70896

70957

71019

71080

71141

71203

71264

71325

~20~

41

46

71984

71995

72055

72116

72176

72236

72297

72367

72417

72477

72537

20

40

47

73135

73195

73254

73314

73373

73432

73491

73551

73610

73669

73728

20

40

48

74314

74373

74431

74489

74548

74606

74664

74722

74780

74838

74896

20

89

49

75471

75528

75585

75642

75700

75756

75813

75870

75927

75984

76041

19

38

50

76604

76661

76717

76772

76828

76884

76940

76996

77051

77107

77162

19

37

51

77715

77769

77824

77879

77934

77988

78043

78098

78152

78206

78261

18

86

52

78801

78855

78908

78962

79016

79069

79122

79176

79_>29

79282

79335

18

86

53

79864

79916

79968

80021

80073

80125

80178

80230

80282

80334

80386

17

35

54

80902

80953

81004

81055

81106

81157

81208

81259

81310

81361

81412

17

34

55

81915

81965

82015

82065

82115

82165

82214

82264

82314

82363

82413

17

83

56

82904

82953

83001

83050

83098

83147

83195

83244

83292

83340

83389

16

32

57

83867

83915

83902

84009

84057

84104

84151

84198

84245

84292

84339

16

31

58

84805

84851

84897

84943

84989

85035

85081

85127

85173

85218

85264

15

31

50

85717

85762

85806

85851

85396

85941

85985

86030

86074

86119

86163

15

30

60

86603

86646

86690

86733

86777

86820

86863

86906

86949

86993

87036

14

29

61

87462

87504

87540

87589

87631

87673

87715

87756

87798

87840

87882

14

28

62

;88295

88336

88377

88417

88458

88499

88539

88580

88620

88661

88701

14

27

63

81H01

89140

89180

89219

89259

89298

89337

89376

89415

89454

89493

13

26

64

89879

89918

89956

89994

90032

90070

90108

90146

90183

90221

90259

13

25

65

90631

90668

90704

90741

90778

90814

90851

90887

90924

90960

90996

12

24

66

91355

91390

91425

91461

91496

91531

915C6

91601

91636

91671

91706

12

23

67

92050

92085

92119

92152

92186

92220

92254

92287

92321

92355

92388

11

23

68

92718

92751

92784

92816

92849

92881

92913

92945

92978

93010

93042

11

22

60

93358

93389

93420

93452

93483

93514

93544

93575

93606

93637

93667

10

21

70

93969

93999

94029

94058

94088

94118

94147

94176

94206

94235

94264

10

20

71

94552

94580

94609

94637

94665

94693

94721

94749

94777

94805

94832

9

19

72

95106

95133

95159

95186

95213

95240

95266

95293

95319

95345

95372

9

18

73

95630

95656

95681

95707

95732

95757

95782

95807

95832

95857

95882

8

17

74

96126

96150

96174

96198

96222

96246

96269

96293

96316

96340

90363

8

16

75

96593

96615

96638

96660

96682

96705

96727

96749

96771

96793

96815

7

15

76

97030

97051

97072

97093

97113

97134

97155

97176

97196

97217

97237

7

14

77

97437

97457

97476

97496

97515

97534

97553

97573

97592

97611

97630

6

13

78

97815

97833

97851

97869

97887

97905

97922

97940

97958

97975

97992

6

12

70

98163

98179

98196

98212

98229

98245

98261

98277

98294

98310

98325

5

11

80

98481

98496

98511

98526

98541

98556

98570

98585

98600

98614

98629

5

10

81

98769

98782

98796

98809

98823

98836

98849

98868

98876

98889

98902

4

9

82

99027

99039

99051

99063

99075

99087

99098

99110

99122

99133

99144

4

8

83

9S255

99265

99276

99286

99297

99307

99317

99327

99337

99347

99357

3

7

84

99452

W461

99470

99479

99488

99497

99506

99514

99523

99531

99540

3

6

85

99619

99627

99635

99642

99649

99657

99664

99671

99678

99685

99692

2

5

86

99756

99762

99768

99774

99780

99786

99792

99797

99803

99808

99813

2

4

87

99863

99867

99872

99876

99881

99885

99889

99893

99897

99901

99905

1

3

88

99939

99942

99945

99948

99951

99953

99956

99959

99961

99963

99966

1

2

80

99985

99986

99988

99989

99990

49991

99993

99994

99995

99995

99996

1

60'

57'

54'

51'

48'

45'

42'

30'

36'

83'

3<y

r

2'

MEAN

NATURAL DIFF.

SINES continued.

MEAN

DIFF.

83'

36'

30'

42*

46 f

48'

6V

54'

67'

6(X

o

1'

2'

71386

71447

71508

71569

71630

71691

71752

71813

71873

71984

44

20

41

72597

72657

72717

72777

72837

72897

72957

73016

73076

73135

43

20

40

73787

73846

73904

73963

74022

74080

74139

74198

74256

74314

42

20

39

74953

75011

75069

75126

75184

75241

75299

75356

75414

75471

41

19

38

76097

76154

76210

76267

76323

76380

76436

76492

76548

76604

40

19

38

77218

77273

77329

77384

77439

77494

77550

77605

77660

77715

39

18

37

78315

78369

78424

78478

78532

78586

78640

78694

78747

78801

38

18

36

79388

79441

79494

79547

79600

79653

79706

79758

79811

79864

37

18

35

80438

80489

80541

80593

80644

80696

80748

80799

80850

80902

36

17

34

81462

81513

81563

81614

81664

81714

81765

81815

81865

81915

35

17

34

82462

82511

82561

82610

82659

82708

82757

82806

82855

82904

34

16

33

83437

83485

83533

83581

83629

83076

83724

83772

83819

83867

33

16

32

84386

84433

84480

84526

84573

84G19

84666

84712

84759

84805

32

16

31

85310

85355

85401

85446

85491

85536

85582

85627

85672

85717

31

15

30

86207

86251

86295

86340

86384

86427

86471

86515

86559

8C603

30

15

29

87079

87121

87164

87207

87250

87292

87335

87377

87420

87462

29

14

28

87923

87965

88006

88048

88089

88130

88172

88213

88254

88295

28

14

28

88741

88782

88822

88862

88002

88942

88981

89021

80001

80101

27

13

27

89532

89571

89610

89649

89087

89726

89704

89803

80841

80879

26

13

26

90296

90334

90371

90408

90446

90483

90520

90557

90504

90631

25

12

26

91032

91068

91104

91140

91176

91212

91248

91283

01319

01355

24

12

24

91741

91775

91810

91845

91879

91914

911)48

91082

92016

02050

23

11

23

92421

92455

92488

92521

92554

92587

92G20

92058

92086

02718

22

11

22

93074

93106

93137

93169

93201

93232

93264

93295

93327

03358

21

10

21

93698

93728

93759

93789

93819

93849

93879

93909

93939

93969

20

10

20

94293

94322

94351

94380

94409

94438

94466

94495

94523

9-1552

19

10

19

94860

94888

94915

94943

94970

94997

95024

95052

95079

95106

18

9

18

95398

95424

95450

95476

95502

95528

95554

95579

95605

95030

17

9

17

95907

95931

95956

95981

96005

96029

96054

96078

06102

96126

16

8

16

96J86

96410

96433

96456

96479

96502

96524

96547

90570

96593

15

8

15

96837

96858

96880

96902

D6923

96945

90966

90987

97008

97030

14

7

14

97257

97278

87298

97318

973,38

97353

97378

07398

07417

97437

13

7

13

97648

97667

97686

97705

97723

97742

97760

97778

07707

97815

12

6

12

98010

98027

98044

98061

98079

98096

98112

98129

98146

98103

11

6

11

98341

98357

98373

98389

98404

98420

98435

98450

98406

98481

10

5

10

98643

98657

98671

98686

98700

98714

98728

98741

98755

98760

9

5

9

98914

98927

98940

98953

98965

98978

98990

99002

99015

99027

8

4

8

99166

99167

99178

99189

99200

99211

99222

99233

99244

09255

7

4

7

99367

99377

99386

99396

99406

99415

99424

99434

99443

99452

6

3

6

99548

99556

99564

99572

99580

99588

99596

99604

99612

99619

5

3

6

99699

09705

99712

99719

99725

99731

99738

99744

99750

99766

4

2

4

99819

99824

99829

99834

99839

99844

99849

99854

99858

99863

3

2

3

99909

99912

99916

99919

99923

99926

99930

99933

99936

99939

2

1

2

99968

99970

99972

99974

99976

99978

99980

99982

99983

99985

1

1

1

99997

99998

99998

99999

99999

99999

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L'OOOOO

00000

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27'

24'

21'

18'

16'

12'

9'

e'

3'

0'

r

2'

MEAN

COSINES. DIFF.

XII.-NATURAL

MEAN

DIFF.

0'

3'

6*

9'

12'

15'

18'

21'

24'

27'

30'

r

2'

ooooo

00087

00175

00262

00349

00436

00524

00611

00698

00785

00873

29

58

01746

01833

01920

02007

02095

02182

02269

02357

02444

02531

02619

29

58

2

03492

03579

03667

03754

03842

03929

04016

04104

04191

04279

04366

29

58

3

05241

05328

05416

05503

05591

05678

05766

05854

05941

06029

06116

29

58

4

06993

07080

07168

07256

07344

07431

07519

07607

07695

07782

07870

29

58

5

08749

08837

08925

09013

09101

09189

09277

09365

09453

09541

09629

29

59

6

10510

10599

10687

10775

10863

10952

11040

11128

11217

11305

11394

29

59

7

12278

12367

12456

12544

12633

12722

12810

12899

12988

18076

13165

30

59

8

14054

14143

14232

14321

14410

14499

14588

14678

14767

14856

14945

30

59

9

15838

15928

16017

16107

16196

16286

16376

164G5

16555

16645

16734

30

60

10

17633

17723

17813

17903

17993

18083

18173

18263

18353

18444

18534

30

60

11

19438

19529

19619

19710

19801

19891

19982

20073

20164

20254

20345

30

60

12

21256

21347

21438

21529

21621

21712

21804

21895

21986

22078

22 169

30

61

13

23087

23179

23271

23363

23155

23547

23G39

23731

23823

23916

2-1008

31

Gl

14

24933

25026

25118

25211

25304

25397

25490

26583

25676

25769

25862

31

62

15

26795

26888

26982

27076

27169

27263

27357

27451

27545

27638

27732

31

62

16

28675

28769

28864

28958

29053

29147

29242

29337

29432

29526

29621

32

G3

17

30573

30669

30764

30860

30955

31051

31147

31242

31338

31434

31530

32

64

18

32492

32588

32685

32782

32878

32975

33072

33169

33266

33363

33460

32

65

19

34433

34530

34628

34726

34824

34922

35020

35118

35216

35314

35412

33

65

20

36397

36496

36595

36694

36793

36892

3G991

37090

37190

37289

37388

33

66

21

38386

38487

38587

38687

38787

38888

38988

39089

39190

39290

39391

34

67

22

40403

40504

40606

40707

40809

40911

41013

41115

41217

41319

41421

34

68

23

42447

42551

42654

42757

428CO

42963

43067

43170

43274

431578

43181

34

09

24

44523

44G27

44732

44837

44942

45047

45152

45257

45362

45467

45573

35

70

25

46631

46737

46843

46950

47056

47163

47270

47377

47483

47590

47698

36

71

26

48773

48881

48989

49098

49206

49315

49423

49532

49640

49749

49858

36

72

27

50953

51063

51173

51283

51393

51503

51614

51724

51835

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52057

37

73

28

53171

53283

53395

53507

53620

53732

53844

53'J57

54070

54183

54296

37

75

29

55431

55545

55659

55774

55888

5G003

56117

56232

5G347

56462

56577

38

77

30

57735

57851

57968

58085

58201

58318

58435

58552

5H670

58787

58905

39

78

31

60086

60205

60324

60443

60562

60681

60801

G0921

61040

61160

61280

40

79

32

62487

62608

62730

62852

G2973

63095

63217

63340

G34G2

63584

63707

41

81

33

64941

65065

65189

65314

65438

65563

65688

65813

659,'J8

GGOG3

66189

42

83

34

67451

67578

67705

67832

67960

68088

68215

68343

68471

G8GOO

68728

43

85

35

70021

70151

70281

70412

70542

70673

70804

70935

71066

71198

71329

44

87

36

72654

72788

72921

73055

73189

73323

73457

73592

73726

73861

73996

45

89

87

75355

75492

75629

75767

75904

76042

76180

76318

76456

76594

76733

46

92

38

78129

78269

78410

78551

78G92

78834

78975

79117

79259

79401

79544

47

95

39

80978

81123

81268

81413

81558

81703

81849

81995

82141

82287

82434

48

97

40

83910

84059

84208

84357

84507

84656

84806

84956

85107

85257

85408

50

99

41

86929

87082

87236

87389

87543

87698

87852

88007

88162

88317

88473

52

103

42

90040

90199

90357

90516

90674

90834

90993

91153

91313

91473

91633

53

107

43

93252

93415

93578

93742

93906

94071

94235

94400

94565

94731

9489G

55

110

44

96569

96738

96907

97076

97246

97416

97686

97756

97927

98098

98270

57

113

60'

57'

54'

51'

48'

45'

42'

39'

36'

33'

30'

r

2'

MEAN

NATURAL DIFF.

TANGENTS.

MEAN

DlFF.

83*

86'

89'

42'

45'

48'

51'

54'

67'

60'

r

2'

00960

01047

01135

01222

01309

01396

01484

01571

01658

01746

89

29

58

02706

02793

02881

02968

03055

03143

03230

03317

03405

03492

88

29

58

04454

04541

04628

04716

04803

04891

04978

05066

05153

05241

87

29

58

OG204

06291

06379

06467

06554

06642

06730

06817

06905

06993

86

29

68

07958

08046

08134

08221

08309

08397

08485

08573

08661

08749

85

29

59

09717

09805

09893

09981

10069

10158

10246

10334

10422

10510

84

29

59

11482

11570

11659

11747

11836

11924

12013

12101

12190

12278

88

29

69

13254

13343

13432

13521

13609

13698

13787

13876

13965

14054

82

30

59

15034

15124

15213

15302

15391

15481

15670

15660

15749

15838

81

30

60

16824

16914

17004

17093

17183

17273

17363

17453

17543

17633

80

30

60

18624

18714

18805

18895

18986

19076

19166

19257

19347

19438

79

30

60

20436

20527

20618

20709

20800

20891

20982

21073

21164

21256

78

30

61

22261

22353

22444

22536

22628

22719

22811

22903

22995

23087

77

31

61

24100

24193

24285

24377

24470

24562

24655

24747

24840

24933

76

31

61

25955

26048

26141

26235

26328

26421

26515

26608

26701

26795

75

31

62

27826

27921

28015

28109

28203

28297

28B91

28486

28580

28675

74

31

63

29716

29811

29906

30001

30097

30192

30287

30382

30478

30673

73

32

63

31626

31722

31818

31914

32010

32106

32203

32299

32396

32492

72

32

64

33557

33654

33751

33848

33945

31043

34140

34238

34335

34433

71

32

65

35510

35608

35707

35805

35904

36002

36101

36199

36298

36397

70

33

66

37488

37588

37687

37787

37887

37986

38086

38186

38286

38386

69

33

67

39492

39593

39694

39795

39896

39997

40098

40200

40301

40403

68

34

67

41524

41626

41728

41831

41933

42036

42139

42242

42345

42447

67

34

68

43585

43689

43793

43897

44001

44105

44210

44314

44418

44523

66

35

69

45678

45784

45889

45995

46101

46206

46312

46418

46525

46631

65

35

71

47805

47912

48019

48127

48234

48342

48450

48557

48665

48773

64

36

71

49967

50076

50185

50295

50404

50514

50623

50733

50843

50953

63

36

73

52168

52279

52390

52501

52613

52724

52836

52947

53059

53171

62

37

76

54409

54522

54635

54748

54862

54975

55089

55203

55317

55431

61

38

76

56693

56808

56923

57039

57155

57271

57386

57503

57619

57735

60

39

77

69022

69140

59258

59376

59494

59612

59730

69849

59967

60086

59

89

79

61400

61520

61641

61761

61882

62003

62124

62245

62366

62487

58

40

81

63830

63953

64076

64199

64322

(54446

64569

64693

64817

64941

57

41

82

66314

66440

66566

66692

66818

60944

67071

67197

67324

67451

56

42

84

68857

68985

69114

69243

69372

69502

69631

69761

69891

70021

55

43

86

71461

71593

71725

71857

71990

72122

72255

72388

72521

72654

54

44

89

74131

74267

74402

74538

74674

74810

74946

75082

75219

75355

53

45

91

76871

77010

77149

77289

77428

77568

77708

77848

77988

78129

52

46

93

79686

79829

79972

80115

80258

80402

80546

80690

80834

80978

51

48

95

82580

82727

82874

83022

83169

83317

83465

83613

83761

83910

50

49

98

85559

85710

85862

86014

86166

86318

86470

86623

86776

86929

49

51

101

88628

88784

88940

89097

89253

89410

89567

89725

89883

90040

48

52

104

91794

91955

92116

92277

92439

92601

92763

92926

93088

93252

47

54

108

95062

95229

95395

95562

95729

95897

96064

96232

96400

96569

46

56

111

98441

98613

98786

98958

99131

99304

99478

99652

99826

I'OOOOO

45

58

115

27'

24'

21'

18'

15'

12'

0'

6'

8'

0'

r

2'

MEAN

COTANGENTS. DOT.

NATURAL

MEAN

DiTF.

o

0'

8*

6'

&

12'

15'

18'

21'

24'

27'

30'

r

2'

45

1-00000

00175

00350

00525

00701

00876

01053

01229

01406

01583

01761

58

117

46

1-03553

03734

03915

04097

04279

04461

04644

04827

05010

05194

05378

61

121

47

1-07237

07425

07613

07801

07990

08179

08369

08559

08749

08940

09131

63

126

48

1-11061

11256

11452

11648

11844

12041

12238

12435

12633

12831

13029

66

131

49

M6037

16240

15443

15647

15851

16056

16261

16466

16672

16878

17085

68

137

50

1-19175

19387

19599

19811

20024

20237

20451

20665

20879

21094

21310

71

142

51

1-23490

23710

23931

24153

24375

24597

24820

25044

25268

25492

25717

74

148

52

1-27994

28225

28456

28687

28919

29152

29385

29618

29853

30087

30323

78

155

53

1-32704

32946

83187

33430

33673

33916

34160

34405

34650

34896

35142

81

162

54

1-37638

37891

38145

38399

38653

38909

39165

39421

39679

39936

40195

85

169

55

1'42816

43080

43347

43614

43881

44149

44418

44688

44958

45229

45501

89

179

56

1-48256

48536

48816

49097

49378

49661

49944

50228

50512

50797

51084

94

189

57

1-53986

54281

54576

54873

55170

55467

55766

56065

56366

56667

5G969

99

198

58

1-60033

60345

60657

60970

61283

61598

61914

62230

62548

62866

63185

105

210

59

1-66428

66757

67088

67419

67752

68085

68419

68754

69091

69428

69766

111

222

60

1-73205

73555

73905

74257

74610

74964

75319

75675

76032

7fi390

76749

118

236

61

1-80405

80777

81150

81524

81899

82276

82654

83033

83413

83794

84177

126

251

62

1-88073

88469

88867

89266

89667

900C9

90472

90876

91282

91 090

92098

134

2G8

63

1-96261

96685

97111

97538

57966

98.196

98828

99261

99695

OOT31

00569

143

287

64

2-05030

05485

05942

06400

06860

07321

07785

08250

08716

09184

09G54

154

307

65

2-14451

14940

15432

15925

16420

16917

17416

17916

18419

18923

19430

1(56

331

66

2-24604

25132

25663

26196

26730

27267

27806

28348

28891

29437

29984

179

358

67

2-35585

36158

3C733

37311

37891

38473

39058

39645

40235

40827

41421

194

388

68

2-47509

48132

48758

49386

50018

50652

61289

51929

52571

53217

53865

69

2-60509

61190

61874

625G1

63252

63945

64612

65342

66046

66752

67462

70

2-74748

75496

76247

77002

77761

78523

79289

80059

80833

81610

82391

71

2-90421

91246

92076

92910

93748

94591

95437

96288

97144

98004

98868

72

3-07768

08685

09606

10532

11464

12400

13341

14288

15240

16197

17159

73

3-27085

28109

29139

30174

31216

32264

33317

34377

35443

36536

37594

74

3-48741

49894

51053

52219

53393

54573

55761

56957

58160

59370

60588

75

3-73205

74512

75828

77152

78485

79827

81177

82537

83906

85284

86671

76

4-01078

02574

04081

05599

07127

08666

10216

11778

13350

14934

16530

77

4-33148

34879

36623

38381

40152

41936

43735

45548

47374

49215

51071

78

4-70463

72490

74534

76595

78673

80769

82882

85013

87162

89330

91516

79

5-14455

16863

19293

21744

24218

26715

29235

31778

34345

36936

39552

80

5-67128

70037

72974

75941

78938

81966

85024

83114

91236

94390

97576

81

6-31375

34961

38587

42253

45961

49710

53503

57339

61219

65144

69116

82

7-11537

16071

20661

25310

30018

34786

39616

44509

49465

54487

69575

83

8-14435

20352

26355

32446

38625

44896

51259

57718

64275

70931

77689

84

9-51436

59490

67680

76009

84482

93101

10-019

10-108

10-199

10-291

10-385

85

11-430

11-546

11-664

11-785

11-909

12-035

12-163

12-295

12-429

12-566

12-706

86

14-301

14-482

14-669

14-860

15-056

15-257

15-464

15-676

16-895

16-119

16-350

87

19-081

19-405

19-740

20-087

20-446

20-819

21-205

21-606

22-022

22-454

22-904

88

28-636

29-371

30-145

30-960

31-821

32-730

33-694

34-715

85-801

36-956

38-188

89

57-290

60-306

63-657

67-402

71-615

76-390

81-847

88-144

95-489

104-17

114-59

60'

57'

54'

51'

48'

45'

42'

39'

36'

33'

30'

1'

2*

MEAN

NATURAL Diif.

39

MEAN

33 1

36*

39'

42'

45'

48'

51'

64'

57'

60'

r

2'

1-01939

02117

02295

02474

02653

02832

03012

03192

03372

03553

44

60

119

1-05562

05747

05932

06117

06303

06489

06676

06862

07049

07237

43

62

124

1-09322

09514

09706

09899

10091

10285

10478

10672

10867

11061

42

64

128

1-13228

13428

13627

13828

14028

14229

14430

14632

14834

15037

41

67

133

1-17292

17600

17708

17916

18125

18334

18544

18754

18964

19176

40

70

139

1-21526

21742

21959

22176

22394

22612

22831

23050

23270

23490

39

73

145

1-25943

26169

26395

26622

26849

27077

27306

27535

27764

27994

38

70

151

1-30558

30795

31031

31269

31507

31745

31984

32224

32464

32704

37

79

159

1-35389

35637

35885

36134

36383

36633

36883

37134

37386

37638

36

83

166

1-40454

40714

40974

41235

41497

41759

42022

42286

42550

42815

35

87

175

1-45773

46046

46320

46595

4C870

47146

47422

47699

47977

48256

34

92

183

1-51370

51658

51946

52235

52525

52816

53107

53400

53693

53986

33

97

193

1-57271

57575

57879

58184

58490

58797

59105

69414

59723

60033

32

102

204

1-63505

63826

64148

64471

64795

65120

65445

65772

66099

66428

31

108

216

1-70106

70446

70787

71129

71473

71817

72163

72509

72857

73205

30

115

229

1-77110

77471

77834

78198

78563

78929

79296

79665

80034

80405

29

122

243

1-84561

81946

85333

85720

86109

86499

86891

87283

87677

88073

28

130

259

1-92508

92920

93332

93746

94162

94579

94997

95417

95838

96261

27

139

277

2-01008

01449

01891

02335

02780

03227

03675

04125

04577

05030

26

148

297

2-10126

10600

11075

11552

12030

12511

12993

13477

13963

14451

25

159

318

2-19938

20449

20961

21475

21992

22510

23030

23553

2-1077

24604

24

172

345

2-30534

31086

31641

32197

32756

33317

33881

34447

35015

35585

23

180

373

2-42019

42618

43220

43825

44433

45043

45655

46270

46888

47509

22

203

405

2-54516

55170

55827

56487

57150

57815

58484

59156

59831

60509

21

2-68175

68892

69612

70335

71062

71792

72626

73263

74004

74748

20

2-83176

83965

84758

85555

86356

87161

87970

88783

89GOO

90421

19

2-99738

oocil

01489

02372

032GO

04152

0~5049

05950

06867

07708

18

3-18127

19100

20079

21063

22053

23048

24049

25055

26067

27085

17

3-38679

39771

40869

41973

43084

44202

45327

46458

47596

48741

16

3*61814

63048

64289

65538

66796

68061

69335

70616

71907

73205

15

3-88068

89474

90890

92316

93751

95196

96G51

98117

99592

01078

14

4-18137

19756

21387

23030

24684

26362

28032

29724

31430

33148

13

4-52941

54826

56726

58641

60572

62518

64480

66458

68452

70463

12

4-93721

95945

98188

00451

02734

05037

07360

09704

12069

14455

11

5-42192

44857

47548

50264

53007

55777

58573

61397

64248

67128

10

6-00797

04051

07340

10664

14023

17419

20851

24321

27829

31376

9

6-73133

77199

81312

85475

89688

93952

98268

02637

07059

11537

8

7-64732

69957

75254

80622

86064

91582

97176

02848

08600

14435

7

8-84551

91520

98598

05789

13093

20516

28058

35724

43M5

51436

6

10-481

10-579

10-678

10-780

10-883

10-988

11-095

11-205

11-316

11-430

5

12-850

12-596

13-146

13-300

13-457

13-617

13-782

13-951

14-124

14-301

4

16-587

16-832

17'084

17-343

17-611

17-886

18-171

18-464

18-768

19-081

3

23'372

23-859

24-368

24-898

25-452

26-031

26-637

27-271

27-937

28-636

2

39-506

40-917

42-433

44-066

45-829

47-740

49-816

52-081

54-561

57-290

1

127-32

143-24

163-70

190-98

229-18

286-48

381-97

572-96

1145-9

Inf.

27'

24'

21'

18'

15'

12'

9'

6'

3'

0'

r

2'

MEA.V

COTANGENTS, DOT.

XIII. -NATURAL

MEAN

DIFF.

0'

3'

6'

9'

12'

15'

18'

2!

24'

27*

30'

r

2'

00000

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2'

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95798

95815

95833

95850

95868

95885

95902

12

66

9-96073

96090

%107

96123

96140

96157

96174

96190

96207

1)6223

96240

6

11

67

9-96103

96419

H6135

96451

1)6467

96483

1)6498

96514

96530

1)65-16

1)6562

5

11

68

9-96717

96732

96747

1)6762

96778

96793

96808

96823

96838

96853

96868

5

10

69

9-97015

97030

97044

97059

97073

97087

97102

97116

97130

97145

97159

5

10

70

9-97299

97312

97326

97340

97353

97367

97381

97394

97408

97421

97435

5

9

71

9-97567

97580

9751)3

97606

97619

97632

97645

97657

97670

97683

97696

4

9

72

9-97821

97833

97815

1)7857

97870

97882

97894

97906

97918

97930

97942

4

8

73

9-98060

98071

98083

9809 1

98106

98117

98129

98140

98151

98162

98174

4

8

74

9-98284

98295

98306

98317

98327

98338

98349

98359

98370

98381

98391

4

7

75

9-98494

98505

98515

98525

98535

98545

98555

98565

98574

98584

98594

3

7

76

9-98690

98700

98709

1KH711)

98728

98737

98746

98756

98765

98774

98783

3

6

77

9-98872

98881

DK890

988D8

98907

98916

98924

98933

98941

98950

98958

3

6

78

9-99040

99048

99056

9906 1

99072

99080

99088

99096

91)104

99112

99119

3

5

79

9-99195

99202

99209

99217

99224

99231

99238

99245

99252

99260

99267

2

6

80

9-99335

99342

99348

99355

99362

99368

99375

99381

99388

99394

99400

2

4

81

9-99162

99468

1)9474

1)1)480

90486

99492

99497

99503

99509

99515

99520

2

4

82

9-99575

99581

99586

995D1

99596

99601

99607

99612

99617

99622

99627

2

3

83

9-99675

99680

9968 1

99689

99693

99698

99702

99707

99711

99716

99720

2

3

84

9-99761

997<>5

99769

99773

99777

99781

99785

99788

99792

99796

99800

1

3

85

9-99834

99838

99841

99844

99847

99851

99854

99857

99860

998G3

99866

1

2

86

9-99894

90897

99899

991)02

99904

99907

99909

99912

99914

99917

99919

1

2

87

9-99940

99912

99944

99946

99948

99950

99952

99954

99955

99957

99959

1

1

88

9-99974

99975

99976

99977

99979

99980

99981

99982

99983

99984

99985

1

89

9-99993

99994

99995

99995

99996

99996

99997

99997

99998

99998

99998

60'

57'

54'

51'

48'

45'

42'

39'

36'

33'

30'

r

2'

MEAN

LOGARITHMIC DIFF.

47

SINES continued.

MEAN

DlFF.

33'

36'

39'

42'

45'

48'

61'

64'

67'

60'

o

r

2'

9-85361

85399

85436

85473

85510

85547

85583

85620

85657

85693

44

12

25

9-86092

86128

86164

86200

86235

86271

86306

86342

86377

86413

43

12

24

9-86798

86832

86867

86902

86936

86970

87005

87039

87073

87107

42

11

23

9-87479

87513

87546

87579

87613

87646

87679

87712

87745

87778

41

11

22

9-88137

88169

88201

88234

88266

88298

88330

88362

88394

88425

40

11

21

9-88772

88803

88834

88865

88896

88927

88958

88989

89020

89050

39

10

21

9-89385

89415

89445

89475

89504

89534

89564

89594

89621

89653

38

10

20

9-89076

90005

90034

90063

90091

90120

90149

90178

90206

90235

37

10

19

9-90546

90574

90602

90630

90657

90685

90713

90741

90768

90796

36

9

19

9-91096

91123

91149

91176

91203

91230

91257

91283

91310

91336

36

9

18

9-91625

91651

91677

91703

91729

91755

91781

91806

91832

91857

34

9

17

9-92136

92161

92186

92211

92235

92260

92285

92310

92335

92359

33

8

17

9-92627

92651

92675

92699

92723

92747

92771

9275

92818

92812

32

8

16

9-93100

93123

93146

93169

93192

93215

93238

93261

93284

93307

31

8

15

9-93554

93577

93599

93621

93643

936G5

93687

93709

93731

93753

30

7

15

9-93991

94012

94034

94055

94076

94098

94119

94140

94161

94182

29

7

14

9'944 JO

91431

94451

94472

94492

94513

94533

94553

94573

94593

28

7

14

9-94813

94832

94852

94871

94891

91911

94930

9494!*

941)69

949H8

27

7

13

9-95198

95217

95236

95254

95273

95292

95310

95329

95318

95361]

26

6

12

9-95567

95585

95603

95621

95639

95657

95674

95692

95710

95728

25

6

12

9-95920

95937

95954

95971

95988

96005

96022

96039

96056

96073

24

6

11

9-96256

96273

96289

9(5305

96322

96338

96354

96370

96387

1)6403

23

5

11

9-96577

96593

96608

961.24

96640

96655

96670

96686

96701

96717

22

5

10

9-96883

96898

96912

96927

96942

96957

96971

961)86

97001

97015

21

6

10

9-97173

97187

97201

97215

97229

97213

97257

97271

97285

97299

20

5

9

9-97448

97461

97475

97488

97501

97515

97528

97541

97554

97567

19

4

9

9-97708

97721

97734

97746

97759

97771

97784

97796

97808

97821

18

4

8

9-97954

97966

97978

97989

98001

98013

98025

98036

9H018

98060

17

4

8

9-98185

98196

98207

98218

98229

98240

98251

98262

98273

9H284

16

4

7

9-98402

98412

98122

98433

98443

98453

98464

98474

98184

98194

15

3

7

9-98604

98614

98623

98633

98643

98652

98662

98671

98681

98690

14

3

6

9-98792

988U1

98810

98819

988. >8

98837

98846

98855

9H864

98872

13

3

6

9-98967

98975

98983

98991

99000

99008

99016

99024

99032

99040

12

8

5

9-99127

99135

99142

99150

99157

99165

99172

99180

99187

1)9195

11

2

5

9-99274

99281

99288

99294

99301

99308

99315

99322

99328

99335

10

2

5

9-99407

99413

99419

99425

99432

99438

99444

99450

99456

99462

9

2

4

9-99526

99532

99537

99543

99548

99554

99559

995(15

99570

99575

8

2

4

9-99632

99637

99642

99647

99651

99656

99661

91)660

99670

99675

7

2

3

9-99724

99728

99733

99737

99741

99745

99749

99753

99757

99761

6

1

3

9-99803

99807

99810

99814

99817

99821

99824

99828

99831

99834

5

1

2

9-99869

99872

99875

99878

99880

99883

99886

99889

99891

99894

4

1

2

9-99921

99923

99926

99928

99930

99932

99934

99936

99938

99940

3

1

j

9-99960

99962

99963

99965

99967

99968

99969

99971

99972

99974

2

1

9-99986

99987

99988

99989

99990

99990

99391

99992

99993

99993

1

1

9-99999

99999

99999

99999

00000

00000

00000

00000

00000

00000

27'

24'

21'

18'

16'

12'

9'

6'

3'

0'

o

1'

2'

MEAN

COSINES. DIFF.

XV.-LOGARITHMIC

HEAX

DlFF.

<y

3'

6'

9'

12'

15'

18'

21'

24'

27'

3CX

I

2'

Inf. Neg.

6-94085

24188

41797

51297

63082

71900

78595

84394

89511

94086

1

8-24192

20312

28332

30-263

32112

3:5880

35590

37229

38809

40334

41807

2

8-54308

55382

56429

57452

58451

50428

60384

61319

62234

63131

64009

3

8-71940

72659

73366

74003

74748

75423

70087

76742

77387

78022

78049

4

8-84464

85006

85540

80069

86591

87106

87616

88120

88618

89111

89598

171

341

5

8-94105

94630

950(10

95486

95008

96325

96739

97150

97556

97959

98358

138

277

6

9-02162

02525

02H85

03212

03507

03048

04297

04043

04987

05328

05060

117

223

7

9-089 J 4

09227

01)537

01)845

101 :>o

10454

10756

11050

11353

110-19

11943

101

202

8

9-14780

15054

15327

15508

15867

1(5135

10401

16665

10028

17100

17450

89

178

8

9-19971

20216

20459

20701

20942

21182

21420

21657

21893

22127

22361

80

159

10

9-24(532

24853

25073

25202

25510

25727

25043

26158

26372

26585

20797

72

144

11

9-28805

20067

21)208

20408

296<)8

29866

30004

30261

30457

30052

30810

66

132

12

9-32747

321)33

a;} 11 9

3U303

Ml 87

33670

33853

34034

34215

31300

34576

61

122

18

9-36336

86509

3608 1

36852

37023

37103

37363

37532

37700

37808

38035

57

113

14

9-39077

39838

39999

40159

40319

40478

40636

40795

40952

41109

41266

53

106

15

9-42805

42957

43108

43258

43108

43558

43707

43855

44004

44151

44209

50

100

16

9-45750

45892

4(10.15

40177

46319

4(5400

40001

46741

4(5881

47021

47100

47

94

17

9-48534

480(19

48804

4MK50

40073

40207

49341

40474

49007

40740

40872

45

89

18

9-51178

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6i y 5

51563

51001

51811)

51946

52073

52200

52320

52452

42

85

10

9-53607

53820

53913

54005

54187

54309

54431

54552

54673

54794

54915

41

81

20

9-56107

56224

50312

50150

50570

50603

50810

50926

57042

57158

57274

39

78

21

9-58418

58531

586 14

5H757

58800

5KOS1

50004

50205

59317

50429

50540

37

75

22

9-60641

00750

00859

00007

01070

61184

01292

01400

61508

01015

61722

36

72

23

9-6-2785

02890

62996

03101

63205

63310

63414

63519

63023

63726

63830

35

70

24

9-64858

61960

65002

65164

65265

65366

65467

65568

65669

65770

65870

34

67

25

9-6C867

66966

67065

67163

67262

67300

67453

67556

67654

67752

67850

33

65

26

9-08818

68914

61)010

01)106

00202

69208

60303

69488

69584

69679

60774

32

64

27

9-70717

70810

70D04

70907

71000

71184

71277

71370

71462

71555

71648

31

62

28

9-72567

72059

72750

72841

72032

73023

73114

73205

73295

73386

73476

30

61

29

9-74375

74465

74554

74G43

74732

74821

74910

74998

75087

75176

75264

30

59

30

9-76144

76231

76319

76406

76493

76580

76668

76754

76841

76028

77015

29

58

31

9-77877

77963

78049

78135

78220

78306

78391

78476

78562

78047

78732

28

57

32

9-79579

79663

71)747

79832

70010

80000

80084

801(58

80251

80335

80419

28

56

33

9-81252

81335

81418

81500

81583

81006

81748

81831

81913

81006

82078

28

55

34

9-82899

82980

83002

83144

83225

83307

83388

83470

83551

83632

83713

27

54

35

9-84523

84603

84684

84764

84845

81025

85006

85086

85166

85247

85327

27

54

36

9-86126

86206

8G285

80365

80145

8652*

80603

80683

8(5762

80842

80921

26

53

37

9-87711

87790

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87048

8H027

88105

88184

88262

88341

88420

88408

26

52

38

9-89281

89359

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89515

89503

80071

89749

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89905

89083

90061

26

52

39

9-90837

90914

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91009

91147

91224

91301

91379

91456

91533

91610

26

52

40

9-92381

92458

92535

92612

92689

92766

92843

92920

92996

93073

93150

26

51

41

9-93916

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94069

94146

94222

91299

94375

94452

94528

94604

94681

25

51

42

9-95444

95520

95596

95672

95748

95825

95901

95977

96053

96129

96205

25

51

43

9-96966

97042

97118

97193

97269

97345

97421

97497

97573

97649

97725

25

51

44

9-98484

98560

98635

98711

98787

98863

98939

99015

99090

99166

99242

25

51

60'

57'

54'

51'

48'

45'

42'

39'

36'

33'

80*

r

2'

MEAN

LOGARITHMIC DIFF.

49

TANGENTS.

MEAN

DIFK.

33'

S&

89'

42'

45'

48'

51'

54'

57'

60'

o

r

2'

7-98225

02004

05481

08700

11600

14500

17133

10016

21004

21192

89

8-43232

44611

45948

47245

48505

49729

50920

52070

53208

54308

88

8-64870

65715

66543

67356

08154

68938

00708

70405

71208

71010

87

8-79266

79875

80476

81008

81653

82230

82799

83301

8301(5

84404

86

8-90080

90557

91029

91495

91957

92414

92806

93313

93750

91105

85

153

306

8-98753

99145

99534

99919

ooijol

00079

07()55

01427

01790

02lT2

84

127

253

9-06002

OG335

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00904

07320

07043

07001

08283

08(500

081)14

83

108

21ti

9-12235

12525

12813

13009

13384

13607

13918

14227

14504

14780

82

94

189

9-17708

17965

18221

18475

18728

18979

19229

10178

10725

10071

81

81

108

9-22593

22824

23054

23283

23510

23737

2J962

24180

24410

21032

80

76

161

9-27008

27218

27427

27635

27842

28049

28254

28459

28002

288(55

79

69

138

9-31040

31233

31425

31610

31800

31990

32185

32373

32501

32717

78

63

127

9-34755

341)33

35111

35288

35164

35040

35815

35080

30103

30.530

77

59

118

9-38202

38308

38534

38G09

38803

39027

30100

30353

39515

3'677

76

65

109

9-41422

41578

41733

41887

42041

42195

42348

42501

42053

42805

75

61

103

9-44446

44592

44738

44884

45029

45174

45319

45403

45000

45750

74

48

97

9*47299

47438

47576

47714

47852

47989

48120

48202

48308

48f>34

73

46

92

9-50004

50136

502(57

503'JS

50529

50659

50789

50919

51048

51178

72

43

87

9-52578

52703

52829

52953

53078

53202

53327

53450

53574

53007

71

41

83

9-55035

55155

55275

55395

55514

55633

55752

55870

55989

50107

70

40

79

9-57389

57504

57619

57734

57849

57963

58077

58191

58304

58118

69

38

70

9-59651

597(52

59872

59983

60093

60203

60313

60422

60532

00011

68

37

73

9-61830

61036

62043

62150

62256

62362

62468

02574

0'2080

0-27H5

67

35

71

9-63934

64037

64140

64243

64346

64449

61552

64654

6 1750

64X58

66

34

69

9-65971

66071

66171

6G271

66371

66470

60570

66609

60708

00807

65

33

66

9-67947

68044

68142

68239

68336

68132

68529

68626

68722

68818

64

32

65

9-09808

69963

70058

70152

70247

70341

70435

70529

70023

70717

63

31

03

9-71740

71833

71925

72017

72109

72201

72293

72381

72476

72507

62

81

61

9-73587

73057

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73837

73927

74017

74107

74106

74280

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61

30

60

9-75353

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75705

75793

76881

75909

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60

29

69

9-77101

77188

77274

77361

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59

29

57

9-78817

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28

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9-83795

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84280

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55

27

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9-85407

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85567

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85727

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85887

85967

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54

27

63

9-87000

87079

87158

87238

87317

87396

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87554

87033

87711

53

26

53

9-88577

88655

88733

88812

88890

88968

89046

89125

89203

89281

52

20

52

9-90138

90216

90294

90371

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90527

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90082

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51

26

52

9-91688

91765

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91919

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92073

92150

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50

26

61

9-93227

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49

26

51

9-94757

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95139

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25

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9-96281

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46

26

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9-99318

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45

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61

27'

24'

21'

18'

15'

12'

&

6'

3' ! O f

c

1'

2'

MEAN

COTANGENTS. Dirr.

LOGARITHMIC

MEAN

Dirr.

a

8'

6'

9'

12'

15'

18'

21'

24'

27'

30'

r

2'

45

10-00000

00076

00152

00227

00303

00379

00455

00531

00606

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51

46

10-01516

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01668

01744

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47

10-03034

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10-10719

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11188

11267

11345

11423

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26

52

53

10-12289

12367

12446

12525

12604

12683

12762

12812

12921

13000

13079

26

58

54

10-13874

13954

14033

14113

14193

14273

14353

14433

14513

14593

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27

58

55

10-15477

15558

15039

15720

15800

15881

15962

16043

16124

16205

16287

27

54

56

10-17101

17183

17205

17347

17429

17511

17593

17675

17757

17839

17922

27

55

67

10-18748

18831

18914

18997

19081

19164

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19414

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28

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58

10-20421

20505

20590

20074

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20844

20928

21013

21098

21183

21268

28

66

59

10-22123

22209

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22381

22467

22553

22639

22726

22812

22899

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29

57

60

10-23856

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24207

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59

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27616

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27799

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28075

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29471

29565

29659

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29848

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10-31182

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11-75808

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82867

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o

60'

57'

54'

61'

48'

45'

42'

39'

36'

33'

30'

r

2

MEAK

LOGARITHMIC Dirr.

MEAN

DIFF.

33'

36'

39'

42'

45'

48'

51'

54'

57'

60'

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2

10-00834

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10-06927

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40

26

51

10-08467

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39

26

62

10-10017

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10173

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38

26

52

10-11580

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11738

11816

11895

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120.V2

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12210

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37

26

52

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36

26

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35

27

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16530

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30

29

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28

30

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10-28445

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28630

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27

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20

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78

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10-88351

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Inf.

27'

24'

21'

18'

15'

12'

9'

6'

3'

:

L

1'

2'

MEAN

COTANGENTS. Dirr.

XVL-LOGARITHMIO

MEAN

DlFK.

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3'

6'

9'

12'

15'

18'

21'

24'

27'

30'

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09482

9

19

37

10-09765

09794

09822

09851

09880

09909

09937

09966

09995

10024

10053

10

19

38

10-10347

J0376

10406

10136

10466

10196

10525

10555

10585

10615

10646

10

20

39

10-10950 ^1980

11011

11042

11073

11104

11135

11166

11197

11228

11259

10

21

40

10-11575

llbCC

11638

11670

11702

11734

11766

11799

11831

11863

11895

11

21

41

10-12222

12255

12288

12321

12354

12387

12421

12454

12487

12521

12554

11

22

42

10-12893

12927

1'2961

12995

13030

13064

13098

13133

13168

13202

13237

11

23

43

10-13587

13623

13658

13694

13729

13765

13800

13836

13872

13908

13944

12

24

44

10-14307

14343

14 380

14417

14453

14490

14527

14564

14601

14639

14676

12

25

o

00'

57'

54'

51'

48'

45'

42'

39'

36'

33'

SO 1

r

2'

i

MEAN

LOGARITHMIC Din.

53

SECANTS.

MEAN

DlFF.

33'

38'

39'

42'

45'

48'

51'

54'

57'

60'

o

r

2'

10-00002

00002

00003

00003

00004

00004

00005

00005

00006

00007

89

10-00016

00017

00018

00019

00020

00021

00023

00024

1)0025

00026

88

i

10-000-13

00045

00046

00048

00050

00052

0005 1

00056

00058

00000

87

1

i

10-00083

00086

00088

00091

00093

00096

00098

00101

00103

00100

86

1

2

10-00137

00140

00143

00146

00149

00153

0015G

00159

00102

00106

85

1

2

10-00204

00208

00212

00215

00219

00223

00227

00231

00235

00239

84

1

3

10-00284

00289

00293

00298

00302

00307

00,111

00310

00320

00325

83

2

3

10-00378

00383

00388

00393

00399

00404

00109

00414

00419

00425

82

2

3

10-00485

00491

00497

00503

00508

00514

00520

00520

00532

00538

81

2

4

10-00006

00612

00619

00625

00632

00638

00645

00052

00058

00605

80

2

4

10-00740

00748

00755

00762

00769

00776

00783

00791

00798

00805

79

2

5

10-00888

00896

00904

00912

00920

00928

00930

00944

00952

00900

78

3

5

10-01050

01059

01067

01076

01084

01093

01102

OHIO

01119

01128

77

3

6

10-01226

01235

01244

01254

012b3

01272

01281

01291

01300

01310

76

3

10-01416

0142C

01435

01445

01455

01465

01475

01485

01495

01500

75

3

7

10-01619

01630

01011

01651

01062

01673

01083

01094

01705

01716

74

4

7

10-01838

01849

018GO

01871

01883

01894

01906

01917

01929

01940

73

4

8

10-02070

02082

02094

02106

02118

02130

02143

02155

02167

02179

72

4

8

10-02317

02330

02343

02355

02308

02381

02394

02407

02420

02433

71

4

9

10-02579

02592

0260G

02619

02033

02647

02060

02674

02088

02701

70

6

9

10-02855

02870

02884

02898

02913

02027

02941

02956

02970

02985

69

6

10

10-03147

03102

03177

03192

03207

03222

03238

03253

03208

03283

68

6

10

10-03454

03470

03486

03502

03517

03533

03549

03505

03581

03597

67

5

11

10-03777

03793

03810

03826

03843

03860

03877

03893

03910

03927

66

6

11

10-04115

04132

04150

04167

04185

04202

04220

04237

04255

04272

65

6

12

10-04469

04487

04506

04524

0-1542

04500

04579

04597

04616

01034

64

G

12

10-04840

04859

04878

04897

04916

04935

04954

04973

04993

05012

63

6

13

10-05227

05247

05260

05286

05300

05326

05346

05306

06380

05407

62

7

13

10-05631

05651

05672

05693

05714

05734

05755

05776

05797

05818

61

7

14

10-06052

06073

06095

06116

OG138

06160

00181

06203

00225

06247

60

7

14

10-06490

06513

06535

00558

OC580

06003

06625

06648

06671

06693

59

8

15

10-06947

06970

06993

07017

07040

07064

07087

07111

07134

07158

58

8

16

10-07421

07445

07470

07494

07518

07543

07507

07592

076 16

07641

57

8

16

10-07914

07940

07965

07990

08015

08041

08000

08092

08117

08143

56

8

17

10-08427

08453

08479

08505

08531

08558

08584

08611

08637

08664

55

9

17

10-08958

08986

09013

09040

09067

09094

09122

09149

09177

09204

54

9

18

10-09510

09538

09566

09595

09623

09051

09080

09708

09737

09765

53

9

19

10-10082

10112

10141

10170

10199

10229

10258

10288

10317

10347

52

10

19

10-10676

10706

10736

10767

10797

10827

10858

10888

109J9

10950

51

10

20

10-11291

11322

11353

11385

11416

11448

11479

11511

11543

11576

60

10

21

10-11928

11960

11993

12025

12058

12091

12123

12156

12189

12222

49

11

22

10-12588

12622

12655

12689

12723

12757

12791

12825

12859

12893

48

11

23

10-13272

13306

13341

13376

13411

13446

13482

13517

13552

13587

47

12

23

10-13980

14016

14052

14088

14124

14161

14197

14234

14270

14307

46

12

24

10-14713

14750

14788

14825

14863

14900

14938

14976

15014

15051

45

13

25

27'

24'

21

18'

15'

12'

9'

6'

3 (

0'

r

2'

MEAN

COSECAJNTS. DIFF

54

LOGARITHMIC

MKAN

Dirr.

(X

3'

6'

9'

12'

15'

18'

21'

24'

27'

3<X

r

2'

45

10-15051

15089

15127

15165

15204

15242

16280

15318

15357

15395

15434

13

25

46

10-15823

15862

15902

15941

15980

16020

1(5060

16099

16139

16179

16219

13

26

47

10-10622

16662

16703

16744

16785

16826

16867

16908

16949

16990

17032

14

27

48

10-17449

17491

17533

17576

17618

17(5(50

17703

17745

17788

17831

17874

14

28

49

10-18306

18349

18393

18437

18481

18525

18569

18613

18657

18701

18746

15

29

50

10-19193

19238

19284

19329

19375

19420

19466

19511

19557

19603

19649

15

30

51

10-20113

20160

20207

20254

20301

20348

20395

20442

20490

20537

20585

16

31

52

10-21066

21114

21163

21212

21201

21309

21358

21408

21457

21506

21555

16

33

53

10-22054

22104

22154

22205

22256

221306

22357

22408

22459

22510

22561

17

34

54

10-23078

23130

23183

23235

23288

23310

23393

23146

23199

23552

23605

18

35

55

10-24141

24195

24249

24304

24358

24413

24467

24522

24577

24632

24687

18

36

56

10-25244

25300

25356

25413

254(59

25526

26583

25640

25697

25754

25811

19

38

57

10-26389

26448

21)506

2(5565

2(5(523

26682

20741

26800

26860

26919

26978

20

39

58

10-27579

27640

27701

277(52

27823

27884

27945

28006

28068

28130

28191

20

40

59

10-28816

28879

28942

29006

29009

29133

29197

29261

29325

29389

29453

21

42

60

10-30103

30169

30235

30301

30H67

30433

30199

30566

30632

30699

30766

22

44

61

10-31443

31511

31580

31619

31717

31787

31856

31925

31994

32064

32134

23

46

62

10-32831)

321)10

32982

3J051

3,5125

3311)7

3J21.U

33342

33414

33487

33559

24

48

63

10-34295

34370

34144

34519

34591

31669

34746

31820

34896

34971

35047

25

60

64

10-35816

35894

35972

36050

36128

36206

36285

36364

36443

36522

36602

26

52

65

10-37405

37487

37568

37650

37732

37814

37896

37979

38061

38144

38227

27

55

66

10- 39069

39154

39239

31)325

39411

31)497

31)583

39069

31)756

39843

;59!);50

28

67

67

10-40812

40902

4091)1

41081

41171

412(51

41352

41443

41533

41(525

41716

30

60

68

10-42(542

42736

42831

4-21)25

43020

43111

4.5210

43305

4:5401

43-11)6

4351)2

32

63

69

10-44567

44666

44765

44804

441)64

45064

45164

45265

45365

45466

45567

33

67

70

10-46695

46699

40804

46908

47014

47119

47225

47331

47437

47544

47650

35

70

71

10-48736

48846

481)57

41)067

49179

49200

49402

49514

49626

49739

49852

37

74

72

10-51002

51119

51236

51353

61471

51589

51708

51827

51946

52066

52186

39

79

73

10-53406

53531

53655

6:5780

53005

51031

54157

54284

54411

54538

54666

42

84

74

10-55966

56099

56231

56365

56498

56(533

56767

56902

57038

57174

57310

45

90

75

10-58700

58842

58984

59127

69270

59414

59558

59703

59848

69994

60140

48

96

76

10-61632

61785

61938

62091

62245

62100

625f)5

62711

(52867

63024

63181

62

103

77

10-64791

64956

65121

(5,V.'7

65453

6>620

65788

05957

66126

66296

66466

56

112

78

10-68212

68391

68570

M7!>0

(58932

61)113

09296

69479

61)664

69849

70034

61

121

79

10-71940

72136

72332

72029

72727

72927

73127

73328

73530

73733

73937

66

133

, 80

10-76033

76248

76465

76683

76902

77122

77343

77565

77789

78013

78239

73

147

81

10-80567

80807

81048

81291

81535

81780

82027

82276

82526

82777

83030

82

164

82

10-85644

85915

86187

8(3461

8G737

H7015

87294

87575

87858

88143

88430

93

185

83

10-91411

91720

92032

92347

92663

92982

93304

93628

93954

94283

94614

107

213

84

10-98077

98439

98804

99172

99544

99918

00296

00678

01063

01451

01843

125

251

85

11-05970

06406

06846

07290

07739

08193

08651

09115

09583

10057

10536

152

303

86

11-15642

16187

16739

17299

17866

18440

19022

19612

20211

20817

21432

87

11-28120

28849

2959 1

30316

31114

31896

32692

33503

31330

35173

36032

88

11-45718

46817

47945

49103

50292

51515

52774

54070

55406

66784

58208

89

11-75814

78042

80390

82872

85505

88307

91304

94522

97998

07777

059T6

60'

6T

54'

51'

48'

45'

42'

39'

36'

33'

30'

1'

2'

LOGARITHMIC

MEAN
Dirr.

SEC Aft T8 continued.

55

MEAN

DlFF.

83'

36'

39'

42'

45'

48'

51'

64'

57'

60'

o

r

2'

10-15472

15511

16550

15589

15627

15666

15705

15745

15784

15823

44

13

26

10-10259

16299

16339

16379

16419

16460

16500

16541

16581

16622

43

13

27

10-17073

17115

17156

17198

17239

17281

17323

17365

17407

17449

42

14

28

10-17916

17959

18002

18045

18089

18132

18175

18219

18262

18306

41

14

29

10' 18790

18834

18879

18924

18968

19013

19058

19103

19148

19193

40

16

30

10-19695

19741

19787

19834

19880

19926

19973

20019

20066

20113

39

16

31

10-20633

20G81

20728

20776

20824

20872

20921

20969

21017

21066

38

16

82

10-21605

21654

21704

21754

21803

21853

21903

21953

22003

22054

37

17

33

10-22613

22664

22715

22767

22819

22870

22922

22974

23026

23078

36

17

34

10-23658

23711

23764

23818

23871

23925

23979

24033

24087

24141

35

18

36

10-24742

24798

24853

24909

24964

25020

25076

25132

25188

25244

34

19

88

10-25868

25926

25983

26041

26099

26157

26215

26273

26331

21)389

33

19

39

10-27038

27098

27157

27217

27277

27337

27398

27458

27518

27579

32

20

40

10-28253

28315

28378

28440

28502

28565

28627

28690

28753

28816

81

21

42

10-29518

29582

29647

29712

29776

29841

29907

29972

30037

30103

30

22

43

10-30833

30900

30968

31035

31103

31171

31238

31306

31375

31443

29

23

45

10-32204

32274

32344

32414

32485

32555

32626

32697

32768

32839

28

24

47

10-33632

33705

33779

33852

33925

33999

31073

34147

34221

34295

27

25

49

10-35123

35200

35276

35353

35429

35506

35583

35661

35738

35816

26

26

51

10-36681

36761

36841

36921

37001

37082

37162

37243

37324

37405

25

27

64

10-38311

38394

38478

38562

38616

38730

38814

38899

38084

39069

24

28

56

10-40017

40105

40192

40280

40368

40457

40545

40(534

407-23

40812

23

29

69

10-41808

41899

41992

42084

42176

42269

42362

42455

42549

42642

22

31

62

10-43689

43785

43882

43979

44077

44174

44272

44370

44468

44667

21

3 '2

65

10-45069

45771

45873

45976

46078

46181

46284

46387

46491

46595

20

31

68

10-47758

47865

47973

48081

48189

48298

48407

48516

48026

4873H

19

36

72

10-49966

50080

50194

50308

50423

50538

50653

60769

50885

5100'2

18

38

77

10-52306

52427

52548

52670

52791

52914

53036

53159

53283

53406

17

41

81

10-54794

54923

55052

55181

55311

55441

55572

55703

55834

651)66

16

43

87

10-57447

57584

57722

67860

57999

58139

58278

58418

58559

58700

15

46

93

10-60287

60434

60582

60730

60879

61029

61179

61330

61481

61632

14

50

99

10-63340

63498

63G58

63818

63978

64140

64302

64464

61627

64791

13

54

107

10-66638

6G810

66982

67156

67330

67505

67681

67857

68034

68212

12

68

116

10-70221

70409

70597

70786

70976

71167

71359

71552

71746

71940

11

63

127

10-74142

74348

74555

74763

74972

75182

76393

75605

75819

76033

10

70

140

10-78466

78694

78924

79155

79387

79620

79855

80091

80328

80567

9

78

155

10-83284

83540

83797

84056

84317

84579

84843

85109

85376

85644

8

87

174

10-88719

89010

89303

89598

89894

90193

90494

90798

91103

91411

7

99

198

10-94948

95285

95624

95966

96310

96658

97008

97361

97717

98077

6

115

230

U'02238

02637

03040

03447

03857

04272

04690

05113

05539

05970

5

137

275

11-11020

11510

12005

12506

13013

13526

14045

14571

15103

15642

4

170

339

11-22057

22690

23333

23985

24647

25320

26003

26697

27403

28120

3

11-36909

37804

38718

39651

40605

41581

42579

43600

44646

45718

2

11-69680

61204

62783

64422

66125

67897

69745

71676

73696

76814

1

12-10491

15607

21406

28100

36018

45709

58203

75812

05915

Inf.

27'

24'

21'

18'

15'

12'

9'

6'

3'

0'

1'

2'

MEAN

COSECANTS. DIFF.

w

a

>v^ -j

^O H^

HH

T

M

^2 t , .5 CN

22S

00

bbbbb

bbbbb

1

632

cs oo t>- to o

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6 4

XXI. EXPONENTIAL AND HYPERBOLIC FUNCTIONS.

X

*

-*

COHII X

SINII x

X

e*

~

COSH x

SINK x

00

1-00000

I'OOOOO

1-00000

00000

50

1*64872

GOG53

1-12763

52110

01

01005

99005

1-00005

01000

51

1-66529

60050

1-13289

53240

02

02021

98020

1-00020

02000

52

1-68203

50452

1-13827

54375

03

03046

97045

1-00045

03000

53

1-69893

58860

1-14377

55516

04

04081

96080

1-00080

04001

54

1-71601

58275

1-14038

56663

05

05127

95123

1-00125

05002

55

1-73325

57695

1-15510

57815

06

0(5184

94177

1-00180

00004

56

1-75067

57121

1-16094

58073

07

07251

93239

1-00245

07006

57

1-76827

56553

1-16690

60137

08

08329

92312

1-00320

08009

58

1-78604

55990

1-17207

01307

09

09417

91393

1-00405

09012

59

1-80399

55433

1-17016

62483

10

10517

90481

1-00500

10017

60

1-82212

54881

1-18547

G3GG5

11

11628

89583

1-00606

11022

61

1-84043

54335

1-10180

04854

12

12750

88692

1-00721

12020

62

1-85893

53704

1-10844

66049

13

13883

87810

1-00846

1 3037

63

1-87761

53259

1-20510

07251

14

15027

86936

1-00982

14046

64

1-89648

52720

1-21189

08459

15

16183

86071

1-01127

15056

65

1-91554

52205

1-21879

69675

16

17351

85214

1-01283

10008

66

1-93479

51G85

1-22582

70897

17

18530

84 366

1-01448

170S2

67

1-95424

51171

1-23207

72120

18

19722

83527

1-01 (.24

18007

'68

1-97388

50662

1-24025

73363

19

20925

82696

1-01810

19115

69

1-99372

50158

1-24765

74G07

20

22140

81873

1-02007

20134

70

2-01375

49G59

1-25517

75858

21

2.1368

81058

1-02213

21155

71

2-03399

40164

1-26282

77117

22

2-1608

80252

1-02430

22178

72

2-05443

48075

1-27050

78384

23

258(50

79453

1-02657

23203

73

2-07."08

48101

1-27850

79659

24

27125

78663

1-02394

24231

74

2-09594

47711

1-28652

80941

25

28403

77880

1-03141

25261

75

2-11700

47237

1-20468

82232

26

2%93

77105

1-03390

2G294

76

2-13828

46767

1-30207

83530

27

30990

76338

1-03607

27320

77

2-15077

46301

1-31139

84838

28

32313

75578

1-03040

28367

78

2-18147

45841

1-31994

86153

29

33643

74826

1-0 1235

29408

79

2-20340

45384

1-32862

87478

30

34986

74082

1-04534

30452

80

2-22554

44033

1-33743

88811

31

36343

73345

1-04844

3141)0

81

2-24791

4448(5

1-34638

90152

32

37713

72615

1-05164

32540

82

2-27050

44043

1-35547

91503

33

39097

71892

1-05495

33602

83

2-20332

43605

1-36468

02863

34

40495

71177

1-05836

34650

84

2-31637

43171

1-37404

94233

35

41907

701GO

1-00188

35710

85

2-33965

42741

1-38353

95612

36

43333

697G8

1*00550

36783

86

2-36316

42316

1-39316

97000

37

44773

G9073

1-06923

37850

87

2-38091

41805

1-40293

98398

38

40228

6838G

1-07307

38921

88

2-41000

41478

1-41284

99806

39

47698

67706

1-07702

39996

89

2-43513

41066

1-42280

1-01224

40

49182

67032

1-08107

41075

90

2-45000

40657

1-43309

1-02652

41

50682

66365

1-08523

42158

91

2-48432

40252

1-44342

1-04090

42

52196

65705

1-08950

43246

92

2-50920

30852

1-45390

1-05539

43

53726

65051

1-0938H

44337

93

2-53451

39455

1-46453

1-06003

44

55271

64404

1-09837

45434

94

2-55998

39063

1-47530

1-08468

45

5G831

637G3

10207

46534

95

2-58571

38674

1-48623

1-00048

46

58.107

63128

10768

47640

96

2-01170

38289

1-40729

1-11440

47

59999

62500

11250

48750

97

2-03794

37008

1-50851

1-12043

48

01607

61878

11743

49865

98

2-60446

37531

1-51988

1-14457

49

1-63232

61263

12247

50985

99

2-69123

37158

1-53141

1-15083

60

1-61872

60653

1-12763

52110

I'OO

2-71828

36788

1-51308

1-17520

i

EXPONENTIAL AND HYPERBOLIC FUNCTIONS continued.

X

-*

COSH x

SiNH X

X

e*

-*

COSH x

SIXH x

I'OO

2-71828

36788

1-54308

1-17520

1-50

4-48169

22313

2-35241

2-12928

roi

2-74560

36422

1-55491

1-19069

1-51

4-52673

22091

2-37382

2-15291

roa

2-77319

36059

1-56689

1-20630

1-52

4-57223

21871

2*39547

2-17676

1-03

2-80107

35701

1-57904

1-22203

1-53

4-61818

21654

2-41736

2-20082

1*04

2-82922

35345

1*59134

1-23788

1-54

4-66459

21438

2-43949

2-22510

t

1-05

2-85765

34994

1-60379

1-2538C

1-55

4-71147

21225

2-46186

2-24961

1-06

2-88637

34646

1-61641

1-26996

1-56

4-75882

21014

2-48 J48

2-27434

1-07

2-91538

34301

1-62919

1-28619

1-57

4-80665

20805

2-50735

2-29930

1-08

2-94-168

33960

1-64214

1-30254

1-58

4-85496

20598

2-53047

2-32449

1-08

2-97427

33622

1-65525

1-31903

1-59

4-90375

20393

2-65384

2-34991

no

3-00417

33287

1-66852

1-33565

1-60

4-95303

20190

2-67746

2-37567

rn

3-03436

32956

1-68196

1-35240

1-61

5-00281

19989

2-60135

2-40146

ria

3-06485

32628

1-69557

1-36929

1-62

5-05309

19790

2-62549

2-42760

ris

3-09566

32303

1-70934

1-38631

1-63

5-103H7

19593

2-64990

2-45397

ri4

3-12677

31982

1-72329

1-40347

1-64

5-15517

19398

2-67457

2-48059

ris

3-15819

31664

1-73741

1-42078

1-66

5-20698

19205

2-69951

2-50746

rie

3-18993

31349

1-75171

1-43822

1-66

5-25931

19014

2-72472

2-53459

ri7

3-22199

31037

1-76618

1-45581

1-67

5-31217

18825

2-75021

2-56196

ris

3-25437

30728

1-78083

1-47355

1-68

5-36556

18637

2-77696

2-58959

ri9

3-28708

30422

1-79565

1-49143

1-69

5-41948

18452

2-80200

2-61748

1*20

3-32012

30119

1-81006

1-50946

1-70

5-47395

18268

2-82832

2-64563

1-21

3-35348

29820

1-82584

1-52764

T71

5-52896

18087

2-85491

2-67405

1-22

3-38719

29523

1-84121

1-54598

1-72

6-58453

17907

2-88180

2-70273

1-23

3-42123

29229

1-85676

1-56447

1*73

6-64065

17728

2-90897

2-73168

1*24

3-45561

28938

1-87250

1-58311

1-74

5-69734

17552

2-93643

2-76091

1-25

3-49034

28650

1-88842

1-60192

1-75

5-75460

17377

2-96419

2-79041

1-28

9-52542

28365

1-90454

1-62088

1-78

5-81244

17204

2-99224

2-82020

1-27

3-56085

280.S3

1-92084

1-64001

1-77

5-87085

17033

3-02059

2-85026

1-28

3-50664

27804

1-93734

1-65930

1-78

5-92986

16864

3-04925

2-83061

1-29

d-63279

27527

1-95403

1-67870

1-79

5-98946

16696

3-07821

2-91125

1-30

3-66930

27253

1-97091

1-69838

1-80

6-04965

16530

3-10747

2-94217

1-31

3-70617

2(5982

1-98800

1-71818

1-81

6-11045

16365

8-13705

2-97340

1'32

3-74342

26714

2-00528

1-73814

1*82

6-17186

16203

3-16694

3-00492

1-33

3-78104

26448

2-02276

1-76828

1'83

6-23389

16041

3' 197 15

4-03674

1-34

3-81904

26185

2-04044

1-77860

1-84

6-29654

15882

3'22768

3*06886

1-35

3-85743

-25924

2-05833

1-79909

1*85

6-35982

15724

3'25853

3-10129

1-36

3-89619

25666

2-07643

1-81977

1'86

i-42374

16567

3-28970

3-13403

1-37

3-93535

25411

2-09473

1-84062

1*87

6-48830

15412

3-32121

3-16709

1-38

3-97490

25158

2-11324

1-86166

1-88

6-55350

15259

3-35305

3-20046

1-39

4-01485

24908

2-13196

1-88289

1-89

6-61937

15107

3-38522

3-23415

T40

4-05520

24660

2-15090

1-90430

1-90

6-68589

14957

3-41773

3-26816

1'41

4-0959G

24414

2-17005

1-92591

T91

6-76309

14808

3'4f)058

d-30250

1-42

4-13712

24171

2-18942

1-94770

1-92

6-82096

14661

3-48378

3-33718

1-43

4-17870

23931

2-20900

1-96970

1-93

6-88951

14515

3-51733

3-37218

1*44

4-22070

23693

2-22881

1-99188

1'94

6-95875

14370

3-55123

3-40752

1-45

4-26311

23457

2-24884

2-01428

1-95

7*02869

14227

3-58548

3-44321

1-46

4-30596

23224

2-26910

2-03686

1-96

7-09933

14086

3-62009

3-47923

1-47

4-34924

22993

2-28958

2-05965

1-97

7-17068

13946

3-65507

3-51561

1*48

4-39295

22764

2-31029

2-08265

1*98

7-24274

13807

3-69011

3-55234

1-49

4-43710

22537

2-33123

2-10586

1'99

7-31553

13670

3-72611

3-58942

1*50

4-48169

22313

2-35241

2-12928

2-00

7-38906

13634

3-76220

3-62686

66

EXPONENTIAL AND HYPERBOLIC FUNCTIONS -continued.

X

e*

-*

Cos u x

SIKH x

x

t*

e-*

COSHZ

SlXH X

2'0

7-38906

13534

3-76220

3-62686

4'0

54-5982

01832

27-3082

27-2899

2*1

8-16617

12246

4-14431

4-02186

41

60-3403

01657

30-1784

30-1619

2*2

0-02501

11080

4'5679l

4-45711

4*2

66-6863

01500

33-3507

33-3357

2*3

9-97418

10026

5-03722

4-93096

4*3

73-6998

01357

36-8567

36-8431

2*4

11-0232

09072

6-66695

6-46623

4*4

81-4509

01228

40-7316

40-7193

2'5

12-1825

08208

6-13229

6-05020

4'5

90-0171

01111

45-0111

45-0030

2*6

18-4637

07427

6-76900

6-69473

4'6

99-4843

01005

49-7472

49-7371

2'7

14-8797

0672L

7-47347

7-40626

4'7

109-947

00910

54-9781

54-9690

2*8

16-4446

06081

8-25273

8-19192

4'8

121-510

00823

60-7593

60-7511

2*9

18-1741

05502

9-11458

9-05956

4*9

134-290

00745

67-1486

67-1412

3'0

20-0855

04979

10-0677

10-0179

6'0

148-413

00674

74-2099

74-2032

3'1

22-1980

04505

1M215

11-0765

51

164-022

00610

82-0140

82-0079

3'2

24-5325

04076

12-2866

12-2459

5*2

181-272

00552

90-6388

90-6333

3'3

27-1126

03688

13-5747

13-6379

5'3

200-337

00499

100-171

100-167

3*4

29-9641

03337

14-9987

14-9654

5*4

221-406

00452

110-705

110-701

3'6

33-1155

03020

16-5728

16-5426

5'5

244-692

00409

122-348

122-344

3'8

36-5982

02732

18-3128

18-2855

5-8

270-426

00370

135-215

135-211

3*7

40-4473

02472

20-2360

20-2113

6'7

298-867

00335

149-435

149-432

3'8

44-7012

02237

22-3618

22-3394

5'8

330-300

00303

165-151

165-148

3'9

49-4021

02024

24-7113

24-6911

5'9

365-037

00274

182-520

182-517

4*0

54-5982

01832

27-3082

27-2899

8'0

403-429

00248

201-716

201-713

XXII.-Loo w '

X

1

2

3

4

5

8

7

8

9

1

43429

47772

52115

56458

60801

65144

69487

73830

78173

82516

2

86859

91202

95545

99888

1-04231

1-08574

1-12917

1-17260

1-21602

1-25945

3

1-30288

1 -34631

1-38974

1-43317

1 '47600

1-62003

1-56346

1-60689

1-05032

1-69375

4

1*73718

1-78061

1-82404

1-86747

1-91090

1-95433

1-99775

2-04118

2-08461

2-12804

6

2-17147

2-21490

2-25833

2-30176

2-34519

2-388G2

2-43205

2-47548

2-61891

2-56234

6

2-60577

2-64920

2-69263

2-73006

2-77948

2-82291

2-86634

2-90977

2-95320

2-99663

7

3-04006

3-08349

3-12692

8-17035

3-21378

3-25721

3-30064

3-34407

3-38760

3-43093

8

3*47436

3-51779

3-56121

3-60464

3-64807

3-69150

3-73493

3-77836

3-82179

3-86522

9

3-90865

3-95208

3-99551

4-03894

4-08237

4-12580

4-16923

4-21266

4-25609

4-29952

10

4-84294

01

02

03

04

05

08

07

08

09

DIFFERENCES

434

809

1303

1737

2171

2C06

3040

3474

3909

XXIII. = (LOO. TAN ^45 + f)) =3-<f- l

*

N

*

u

u

*

u

o

o

o

o-o

00000

22'5

40320

45-0

88137

67'6

1*61489

0'5

00873

23-0

41206

45-5

89377

68*0

1-63794

ro

01745

23*5

42216

46'0

90628

68-6

1-66148

T5

02618

24-0

43169

46'5

91890

69-0

1-68557

2-0

03491

24*5

44127

47-0

93163

69*5

1*71020

2'5

043G5

25'0

45088

47'5

94449

70-0

1-73542

3-0

05238

25'5

46052

48'0

95747

70-5

1*76124

3'5

06112

26-0

47021

48'5

97057

71'0

1-78771

4'0

OG987

26'5

47994

49*0

98381

71'5

1-81486

4'5

07862

27'0

48972

49*5

99718

72-0

1-84273

5*0

08738

27'5

49953

50'0

1-01068

72*5

1-87136

6'5

09614

28'0

50939

50-5

02133

73-0

1'90079

6-0

10491

28'5

51930

51'0

03812

73'5

1-93107

6*5

11369

29*0

52925

51*5

05207

74*0

1-96226

7'0

12248

29'5

53925

52'0

06616

74*5

1-99441

7'5

13128

30*0

54931

62-5

08042

75'0

2-02769

8-0

14008

30*5

55941

63*0

09483

75'5

2-06187

8'5

14890

31-0

56956

53*5

10942

76-0

2-09732

9'0

15773

31-5

57977

64-0

12418

76'5

2-13404

9'5

16657

32-0

59003

54-5

13911

77-0

2-17212

10-0

17543

32-5

60035

65-0

15423

77'5

2-21167

10-5

18429

33-0

61073

55-5

16954

78*0

2-25280

iro

19318

33-5

62116

56-0

18505

78'5

2-29566

11-6

20207

34-0

63166

56-5

20076

79-0

2-34040

12'0

21099

34-5

64222

57-0

21667

79'5

2-38719

12-5

21992

35-0

65284

67-6

23281

80'0

2-43625

13-0

22886

35'5

66352

58'0

24916

80'5

2-48779

13'5

23783

36-0

67428

68'5

1-20574

81'0

2-64209

14'0

24681

36-5

68510

69-0

1-28257

81-5

2-69947

14'5

25582

37-0

69599

59-5

1-29963

82'0

2-66031

15-0

26484

37'5

70695

80-0

1-31696

82'5

2-72501

15-5

27389

38'0

71799

60-5

1-33454

83'0

2-79422

16-0

28295

38'5

72910

61-0

1-35210

83'5

2-86850

16'5

29204

39-0

74029

61-6

1-37055

84-0

2-94870

17'0

30116

39'5

75156

62-0

1-38899

84'6

3-03586

17'5

31030

40'0

76291

62'5

1-40773

86-0

3-13130

18-0

31946

40*5

77434

63-0

1-42079

855

3-23678

18*5

32865

41*0

78586

63-5

1-44618

86-0

3-35467

19'0

33786

41-6

79747

64-0

1-46591

86*5

3-48830

19-5

34711

42*0

80917

64'5

1-48600

87'0

3-64253

20-0

35638

42'5

82096

65-0

1-50645

87*5

3-82492

20-5

36568

43-0

83284

65'5

1-52730

88-0

4-04813

21-0

37501

43'5

84482

66*0

1-54855

88*5

4-33585

21'5

38438

44-0

85690

66*5

1-57022

89*0

4-74135

220

39377

44*5

86909

67-0

1*69232

89'5

5-43451

90*0

Inf.

6S

XXIV. ELLIPTIC

*

=

= 50

= 10

= 15

= 20

= 25

= 30

= 35

= 40 C

= 45

1

01745

01745

01745

01745

01745

01745

01745

01745

01745

01745

a

03491

03491

03491

03491

03491

03491

03491

03491

03491

03491

3

05230

05236

05230

05236

05236

05236

05237

05237

05237

05237

4

00981

06981

00981

06982

06982

06982

OC!)83

06983

06984

06984

5

08727

08727

08727

08727

08728

08729

08729

08730

08731

08732

6

104/2

10472

10473

10473

10474

10475

10477

10478

10480

10482

7

12217

12218

12218

12219

12221

12223

12225

12227

12230

12233

8

13963

13963

13904

1390G

13968

13971

13974

13978

13981

13985

9

16708

15708

15710

15712

15715

15719

15724

15729

15735

15740

10

17453

17454

17456

17459

17464

17409

17475

17482

17490

17498

11

19199

19200

19202

19206

19212

19220

19228

19237

19247

19258

12

20944

20945

"2094!)

20954

20962

201)71

20982

20994

21007

21021

13

22G8!)

22691

220D5

22702

22712

22724

22738

22753

22770

22787

14

24435

24436

24442

24451

24403

24478

24495

24514

24535

24556

15

2G180

26182

26189

26200

20215

26233

26254

26278

20303

26330

16

27925

27928

27936

27949

27967

27989

28015

28044

28075

28107

17

29671

29674

29081

29699

29721

21)748

29779

29813

29850

29889

18

31416

31420

3i4:u

31450

31475

31507

31544

31585

31629

31675

10

33161

33166

33179

33201

33231

33208

33312

33300

33412

3.'I4 66

20

34907

34912

34927

34953

34988

35031

35082

35138

35199

35262

21

36652

36658

36676

36706

30746

30796

36855

36920

36990

37063

22

38397

38404

38425

38459

38505

38563

38630

38705

38786

38871

23

40143

40151

40174

40213

40200

40331

40408

40494

.40587

40683

24

41888

41897

41924

41908

42027

42102

421H9

42287

42392

42503

25

43633

43643

43674

43723

43791

43875

43973

44081

44203

44328

26

45379

45390

45424

45479

45555

45650

45761

45885

46020

46161

27

47124

47137

47174

47230

47321

47427

47551

47090

47841

48000

28

48869

48883

48925

48994

49089

49207

49345

49500

49069

49840

29

50615

50630

50677

50753

50858

50988

51142

51315

51503

51700

30

52360

52377

62428

52513

52628

52773

52943

53134

53343

53562

31

54105

54124

'54181

54273

54401

54560

54747

54959

56189

55432

32

55851

55871

55933

50035

56175

56349

56555

56788

57042

57310

33

57596

57619

57686

57797

57950

58141

58367

58623

58902

69197

34

59341

69366

59439

59501

59727

59936

60183

00463

60769

61093

35

61087

61113

61193

61325

61506

61734

62003

62308

62643

62998

36

62832

62861

62948

63090

63287

63534

63827

64159

64524

64912

37

64577

04609

64702

64857

65070

65337

65655

66016

66413

66836

38

66323

66356

66457

60624

6G854

67144

67487

07879

68309

68769

39

C8068

68104

68213

68393

68641

68953

69324

69747

70214

70713

40

69813

69852

69969

70162

70429

70765

71165

71622

72126

72667

41

71558

71600

71726

71933

72219

72680

73010

73502

74047

74632

42

73304

73349

73483

73704

74011

74398

74860

75389

75976

76608

43

76049

75097

75240

75477

75805

76219

76714

77282

77914

78594

44

76794

76846

70998

77251

77600

78013

78573

79182

79860

80592

45

78540

78594

78756

79026

79398

79871

80437

81088

81815

82302

FUNCTIONS. F(0,

69

<9 = 50

= 65

</ = 60

4 = 65

= 70

= 75

* = 80

= 85

o = 9<)o

*

01745

01745

01745

01745

01745

01745

01745

01745

01745

1

03491

03491

03491

03491

03491

03491

03491

03491

03491

2

05237

05238

05238

05238

05238

05238

05238

05238

05*238

3

06985

06985

06986

00986

06986

0(5987

06987

00987

00987

4

08733

08734

08735

08736

08736

08737

08737

08738

08738

5

10483

10485

10486

10488

10489

10190

10491

10491

10491

6

12235

12238

12240

12242

12244

122 10

12247

12218

12218

7

13989

13993

13997

14000

14003

J 100.")

14007

14008

14008

8

15746

15751

15757

15761

15705

15709

15771

15772

15773

9

17605

17513

17520

17526

17532

17537

17540

17542

17513

10

19268

19278

19288

19296

19301

19310

19314

19317

19318

11

21034

21047

21059

"21071

21080

2 loss

21094

21098

21099

12

22801

22821

22K30

22851

22803

22873

22HHO

22885

2'286

13

24578

24599

2401 8

24030

24052

21001

24074

24(580

24(581

14

26356

26382

26406

20428

20448

20403

26475

20482

20484

16

28139

28171

28200

28227

28251

28270

28284

28293

28295

16

29927

29.%5

30001

MOM

30002

300H, r >

30102

30112

30110

17

31721

31706

31809

3184H

31881

31900

31929

31942

31946

18

33520

33574

33624

33070

33710

33712

33706

33781

33786

19

35326

35388

35447

35501

35548

3558(5

35615

35032

36638

20

37137

'.37210

37279

37342

373%'

3744 1

37474

37494

37501

21

38956

31KHO

39119

39192

39255

39307

39346

393(59

39377

22

40782

40878

40909

41053

41120

411X0

41230

41257

41200

23

42614

42724

42829

42925

43008

43077

43128

43159

43109

24

44455

44580

44099

44808

44904

44982

45040

45075

45088

25

46304

46445

46580

46704

46812

46901

46967

47008

47021

26

48161

48320

48472

48012

48735

48835

48910

48956

48972

27

50027

50206

30377

50534

50072

507H5

50870

50922

50939

28

51902

52102

52293

52470

52024

52752

52847

52905

52925

29

53787

54009

64223

54420

54593

54736

54843

54908

54931

30

55681

55928

56166

56386

50579

56739

56858

56931

56966

31

57586

57860

58123

58367

58582

58700

58893

58975

59003

32

59501

59803

60095

60305

60604

GOS02

00950

61042

61073

33

61427

61760

02082

62381

62646

62865

63029

03131

63166

34

63364

63730

64085

64415

64707

64950

65132

65245

6*284

35

65313

65715

66104

66468

66790

67058

67260

67386

67428

36

67273

67713

68141

G8540

68895

69191

69414

69562

69599

37

69246

69727

70195

70633

71023

71349

71594

71747

71799

38

71232

71756

72267

72746

73175

73. r )33

73804

73972

740'29

39

73231

73801

74358

74882

'75352

75745

76043

76228

76291

40

76243

^75862

76469

77041

77555

77987

78313

78517

78586

41

77269

77940

78600

79224

79786

80258

80617

80841

80917

42

79308

80035

80752

81432

82045

82562

82954

83200

83284

43

81362

82149

82926

83665

84333

84898

85329

85598

85690

44

83431

84281

85122

85925

86053

87270

87741

88037

88137

45

i

ELLIPTIC FUNCTIONS,

*

=

= 5

3 = 10

= 15

= 20

= 25

= 30

= 35

= 40

= 450

46

80285

80343

80515

80801

81198

81701

82305

83001

83779

84623

47

82030

82092

82275

82578

82999

83535

84178

8-1920

85752

86656

48

83776

83841

84035

84356

84803

85371

86065

86846

87734

88701

49

85521

85590

85795

86135

86609

87211

87937

88779

89725

90759

60

872G6

87339

87556

87915

88416

89054

89825

90719

91725

92829

51

89012

89088

89317

896.07

D0226

90901

91716

92CG5

93735

94912

52

90757

90838

91078

91479

92037

92750

93613

94618

95755

97007

53

92502

92587

92841

93262

93850

94603

95514

96578

97784

99115

54

94248

91337

94003

95047

95666

96458

97420

98545

99822

1-01237

55

95993

96086

96366

96832

97483

98317

99331

1-00519

1-01871

1-03371

56

97738

97836

98130

98618

99302

1-00179

1-01247

1-02499

1-03928

1-05519

57

99484

99586

99894

1-00406

1-01123

1-02044

1-03167

1-04487

1-05996

1-07680

58

1 '01 229

1-01336

1-01658

1-02194

1-02946

1-03912

1-05092

1-06481

1-08073

1-09854

59

1-02974

1-03086

1-03423

1-03981

1-04770

1-05783

1-07021

1-08482

1-10159

1-12042

60

1*04720

1-04837

1-05188

1-05774

1-06597

1-07657

1-08955

1-10190

1-12256

1-14243

61

1-06465

1-06587

1-06954

1-07566

1-08425

1-09534

1-10894

1-12504

1-14361

1-16457

62

1-08210

1-OH338

1-08720

1-09358

1-10255

1-11414

1-12837

1-14525

1-16476

1-18685

63

! 09956

1-10088

10486

1-11151

1-12087

1-13296

1-14784

1-16552

1-18601

1-20926

64

1-11701

1-118:59

12253

1-12945

1-13920

1-15182

1-16735

1-18586

1-20735

1-23180

65

1-13446

1-13590

14020

1-14740

1-15755

1-17070

1-18691

1-20626

1-22877

1-25447

66

1-15192

1-15340

15787

1-16536

1-17592

1-18961

1-20651

1-22672

1-25029

1-27727

67

1*16937

1-17091

17555

1-18333

1- 194I50

1-20854

1-22615

1-24724

1-27190

1-30020

68

1-18(582

1-18842

19324

1-20130

1-212(59

1-22750

1-24583

1-26782

l-29,'559

1-32325

69

1-20428

1-20593

1-21002

1-21928

1-23110

1-24648

1-26555

1-28846

1-31537

1-34642

70

1-22173

1-22345

1-22861

1-23727

1-24953

1-26548

1-28530

1-30915

1-33723

1-36972

71

1-23918

1-24096

1-24630

1-25527

1-26796

1-28451

1-30509

1-32990

1-35917

1-39313

72

1 '26664

1-25847

1-26400

1-27328

1-28641

1-30356

1-32491

1-35070

1-38118

1-41666

73

1-27409

1-27599

1-28169

1-29129

1-30488

1-32263

1-34477

1-371,* 5

1-40328

44030

74

1-29154

1-29350

1-29939

1-30930

1-32335

1-34172

1-36466

1-39244

1-42544

46404

75

1-3U900

1-31102

1-31710

1-32733

1-34184

1-36083

1-38457

1-41339

1-44767

1-48788

76

1-32645

1-32853

1-33480

1-31535

1-36034

1-37996

1-40452

1-43437

1-46997

61183

77

1-34390

1-34605

1-35251

1-36339

1-37884

1-39911

1-42449

1-45540

1-49232

53586

78

1-3G136

1-30356

1-37022

1-38143

1-39736

1-41827

1-44449

1-47647

1-51474

55999

79

1-37881

1-38108

1-38793

1-39947

1-41588

1-43744

1-46151

1-49757

1-53721

1-58419

80

1-39626

1-39860

1-40565

1-41752

1-43442

1-456G3

1-48455

1-51870

1-55973

1-60848

81

1-41372

1-41612

1-42336

1-43557

1-45296

1-47583

1-50462

1-53987

1-58230

1-63283

82

1-43117

1-43364

1-41108

1-45362

1-47150

1-49504

1-52470

1-56106

1-60491

1-65725

83

1-44862

1-45115

1-45879

1-47168

1-49005

1-51426

1-54479

1-58228

1-62756

1-68172

84

1-4G608

1-46867

1-47651

1-48974

1-50861

1-53350

1-56490

1-60352

1-65024

1-70(525

85

1-48353

1*48619

1-49423

1-50781

1-52717

1-65273

1-58503

1-62478

1-67295

1-73082

86

1-50098

1-50371

1-51195

1-52587

1-54574

1-57198

1'60516

1-64605

1*69569

1-75542

87

1-51844

1-52123

1-52968

1-54394

1-56431

1-59123

1-62530

1-66734

1*71844

1-78006

88

1-53589

1-53875

1-54740

1-56200

1-58288

1-61048

1-64545

1-68864

1-74121

1-80472

89

1-65334

1-55627

1-56512

1-58007

1-60145

1-62974

1-66560

1'70994

1*76399

1-82939

90

1-57080

1-57379

1-58284

1-59814

1-62003

1-64900

1-68575

1*73125

1*78677

1-85407

0, \$) continued.

* = 50

= 55

= 60

= 65

0=70

= 75

<? = 80

' = 85

o = 90

+

'85515

86431

87342

88213

89005

89678

90193

90517

90628

46

870 14

88601

89585

90529

91390

92124

92(587

93042

931G3

47

89729

90791

91853

92875

93811

94610

95226

95614

95747

48

91860

93001

94146

95252

96267

97139

97810

98235

98381

49

94008

95232

96465

97660

98762

99711

1-00444

1-00909

1-01.068

50

96171

C7<*84

98811

1-00102

1-01297

1-02329

1-03129

1-03638

1-08812

61

98352

99709

1-01185

1-02578

1-03872

1-04995

1-05868

1-06425

1-00010

52

1-00550

1-02055

1-03587

1 '05089

1-06491

1-07711

1-08665

1-09274

1-09183

53

1-02765

1-04374

1-06018

1-07637

1-09155

1-10481

1-11521

1-12188

1-12118

54

1-04998

1-06716

1-08479

1-10223

1-11865

1-13307

1-14442

1-15171

1-15423

55

1-07248

1-09082

1-10971

1-12848

1-14624

1-16190

1-17430

1-18229

1-18505

56

09517

1-11472

1-13494

1-15513

1-17433

19136

1-20488

1-21364

1-21067

57

11803

1-13886

! 16050

1-18220

1-20295

22145

1-23G23

1-24582

1-24916

58

14108

1-16325

P1H638

1-20970

1-23212

25223

1-26837

1-27890

1-28267

59

16432

1-18788

1-21254

1-23764

1-26186

28371

1-30135

1-31292

1-31696

60

18773

1-21277

1-23916

1-20604

1-29219

31594

1-33524

1-34795

1-35240

61

21134

1-23792

1-26606

1-2IM90

1-32314

34896

1-37008

1-38407

1-38899

62

23513

1-26332

1-29332

1-32425

1-35473

38281

1-40594

1-42135

1-42079

63

25910

1-28898

1-32094

1-35409

1-38099

41753

1-44288

1-45989

1-46591

64

1-28326

1-31491

1-34893

1-38443

1-41994

45316

1-48098

1-49977

1-50645

65

1-30760

1-34109

1-37728

1-41529

1-45360

48976

1-52031

1-54112

1-54855

66

1-33212

1-36753

1-40600

1-44668

1-48800

52738

1-56096

1-58404

1-59232

67

1-35683

39423

1-43510

1-47860

1-52317

5(1006

1-C0303

1-62868

1-63794

68

1-38171

42119

1-46457

1-51107

1-55913

(50586

1-64661

1-67518

1-08557

69

1-40677

1-44840

1-49441

1-54410

1-59591

64684

1-69181

1-72372

1-73542

70

1-43200

47587

1-52463

1-57768

1-63352

68905

1-73877

1-77450

1-78771

71

1-45739

50359

1-55522

1-61182

1-67198

73256

1-78759

1-82774

1-84273

72

1-48296

53155

1-58618

1-64653

1-71132

77743

1-83844

1-88370

1-90079

73

1-50867

55974

1-01750

1-68180

1-75155

82371

1-89146

1-94267

1-90220

74

1-53455

58817

1-64918

1-71763

1-79269

1-87145

1-94682

2-00499

2-02759

75

1-56056

1-61682

1-68120

1-75401

1-83473

1-92073

2-00470

2-07106

2-09732

76

1-58672

1-64569

1-71356

1-79094

1-87768

1-97157

2-06529

2-14136

2-17212

77

1-61302

1-67476

1-74625

1-82840

1-92154

2-02403

2-12878

2-21044

2-25280

78

1-63943

1-70403

1-779-24

1-86637

1-96630

2-07813

2-19538

2-29694

2-J4040

79

1-66597

1-73347

1-81253

1-90484

2-01193

2-13390

2-26527

2-38365

2-43625

80

1-69261

76309

1-84609

1-94377

2-05840

2-19131

2-33866

2-47748

2-51209

81

1-71935

79286

1-87'.91

1-98313

2-10568

2-25035

2-41569

2-57954

2-66031

62

1-74618

82278

1-91395

2-02290

2-15371

2-31097

2-49648

2-69109

2-79422

83

1-77309

85281

1-94821

2-06.J03

2-20244

2-37309

2-58105

2-81362

2-94870

84

1*80006

88296

1-98264

2-10348

2-25178

2-43658

2-66935

2-94869

3-13130

85

1-82710

1-91320

2-01723

2-14421

2-30166

2-50129

2-76116

3-09782

3-35467

86

1-85418

1-94351

2-05194

2-18515

2-35198

2-56703

2-85612

3- 201 98

3-04253

87

1-88129

1-97388

2-08674

2-2-2627

2-40265

2- (53357

2-95366

3-44116

4-04813

88

1-90843

2-00429

2-12161

2-26750

2-45354

2-70068

3-05304

3-63279

4-74135

89

1*93558

2-03472

2-15652

2-30879

2-50455

2-76806

3-15339

3-83174

Inf.

90

XXV.-ELLIPTIO

*

tf =

e = b

= 10

= 15

= 20

= 25

= 30.

= 35

= 40

= 46

1

01745

01745

01745

01745

01745

01745

01745

01745

01745

01745

2

03491

03491

03491

03491

03491

03491

03490

03490

03490

03490

3

0523G

05236

05236

05236

05236

05236

05235

05235

05235

05235

4

06981

06981

06981

06981

06981

OG980

OG980

06979

06979

06078

5

08727

08727

08726

08726

08725

08725

08724

08723

08722

08721

6

10472

10472

10471

10471

10470

10469

10467

10466

10464

10462

7

12217

12217

12216

12215

12214

12212

12210

12207

12205

12202

8

13963

13962

13961

13960

13957

13955

13951

13948

13944

13940

9

15708

15707

15706

15704

15700

15G9G

15692

15G87

15681

15676

10

17453

17453

17451

17447

17443

17438

17431

17424

17417

17409

11

19199

19198

19195

191JU

19185

19178

191G9

19160

19150

19140

12

20044

20943

20939

20931

20926

20917

20906

20894

20881

20868

13

22(589

22688

22683

22G76

22667

22G55

22641

22G26

22609

22593

14

244:15

24433

24427

24419

24406

24392

24374

24355

24335

24314

15

2G180

2G178

26171

2G160

26145

26127

26106

26083

26058

26032

16

27925

27923

27914

27901

27883

27861

27836

27807

27777

27746

17

29671

29(i67

29058

29642

29620

29594

29563

29529

29493

29455

18

31416

31412

31401

31382

31357

31325

31289

31248

31205

31161

19

33161

33157

33143

33121

33092

33055

33012

32965

32914

32862

20

34907

34901

34886

34860

34825

34783

34733

34678

34619

34558

21

36652

36646

36628

36598

36558

36509

36451

36387

36319

36249

22

88397

38390

38370

3H33G

3821)0

38233

38167

38093

38015

37934

23

40143

40135

40111

40073

40020

39955

39880

39796

39707

39614

24

41888

41879

41852

41809

41749

41676

41590

41496

41394

41289

25

43633

43G23

43593

43514

43477

43394

43298

43191

43076

42958

26

45379

45367

45333

45278

45203

45110

45002

44882

44753

44620

27

47124

47111

47074

47012

46928

46824

46703

46569

46425

4627G

28

48869

48855

48813

48745

48651

48536

48402

48252

48092

47926

29

50615

50599

50553

50477

50373

50245

50097

49931

49753

49569

30

62360

52343

52292

52208

52094

51953

51788

51605

51409

51205

31

54105

54086

54030

53938

53813

53657

53476

53275

53059

52834

32

55851

55830

55768

55667

55530

553GO

55161

54940

54703

54456

33

57596

57573

57506

57396

57245

57059

56842

56600

56341

56070

84

59341

59317

59243

59123

58959

58756

58520

58256

57972

57677

35

01087

61060

60980

60850

60672

60451

60194

69907

59598

59276

36

62832

62803

62716

62575

62382

62143

61864

61552

61217

60868

37

64577

64546

64452

64300

64091

63832

63530

63193

G2830

62451

38

66323

66289

66188

66023

65798

65519

65193

64828

64436

64027

39

68068

68081

67923

67746

67503

67203

66851

66459

66035

65594

40

69813

69774

69658

69467

69207

68884

68506

68081

67628

67153

41

71553

71517

71392

71188

70909

70562

70157

69703

69214

68703

42

73304

73259

73126

72907

72609

72238

71804

71318

70793

70245

43

75049

75001

74859

74626

74307

73910

73446

72927

72365

71778

44

76794

76744

76592

76343

76003

75580

75085

74530

73931

73303

46

78540

78486

78324

78059

77697

77247

76720

76128

75489

74819

FUNCTIONS. E(0,</).

73

= 60

0=65

= 60

= 65

= 70

= 75

= 80

= 86

= 90

*

01745

01745

01745

01745

01745

01745

01745

01745

01745

1

03490

03490

03490

03490

03490

03490

03490

03490

03490

2

05235

05234

05234

05'234

05234

05234

05234

05^234

05234

3

06978

06978

06977

06977

OG976

06976

00976

06076

OG976

4

08720

08719

08718

08718

08717

08716

08716

08716

08716

5

10461

10459

10458

104 50.

10155

10454

10453

10453

10453

6

12199

12197

12195

12192

12190

12189

12188

12187

12187

7

13936

1S932

13929

13925

131)23

13WO

131)19

13918

'13917

8

16670

166G5

15(560

16656

15(551

16618

15(545

15644

15643

9

17401

17394

17387

17381

17375

17371

173G7

173G5

17365

10

19180

19120

19110

19102

19095

19089

19084

19082

19081

11

20855

20842

20830

20819

20809

20801

20796

20792

20791

12

22576

22559

22544

22530

22518

22508

22501

22-197

2*195

13

24293

24272

24253

2423U

24221

24209

24200

24194

24192

14

26006

25981

25957

25936

25917

25902

25891

25884

26882

15

27714

27684

27655

27G29

27G06

27588

27575

27567

27564

16

29418

29381

29347

29315

29288

292G7

29250

29241

29237

17

31116

31073

31032

30995

30963

30937

30!) 17

30JKJG

30902

18

82809

32758

32710

32666

32629

32598

32575

32561

32557

19

84496

34437

34381

34330

34286

34250

34224

34207

84202

20

36178

36109

3C044

35985

35934

35892

35862

35843

35837

21

87853

37773

37699

37631

37572

37525

37490

37468

37461

22

39521

39431

39345

39268

39201

39146

39106

3 ( .)()8l

39073

23

41183

41080

40983

40895

40819

40757

40711

40G83

40674

24

42838

42722

42612

42513

42426

42356

42304

42273

42262

25

44486

44355

44232

44120

44023

43944

43885

438-19

43837

26

46126

45980

45842

45716

45607

45518

45453

45413

45399

27

47759

47695

47441

47301

47180

4 70S I

47007

46962

4G947

28

49383

49202

49031

48875

48740

48G29

48548

48498

48481

29

61000

50799

50609

50437

60287

50165

50074

50019

50000

30

52608

52386

52177

51986

51821

51686

51586

61525

61504

31

54207

53964

53733

53524

53341

53193

53082

53015

52992

32

55798

55531

55278

55048

54848

54684

54563

54489

64464

33

67379

57087

56811

56559

5G340

56161

56028

55947

55919

34

58952

58634

58332

58057

67818

57622

57477

57388

67368

35

60515

60169

59841

59541

59280

590G7

58909

58811

58779

36

62068

61693

61337

61011

60727

60495

60323

60217

60182

37

63612

63206

62820

62467

62159

61907

61720

61605

61566

38

65146

64707

64290

63908

63574

63302

63099

62974

62932

39

66671

66197

65746

65334

64974

64679

64459

64324

64279

40

68185

67675

67189

66745

66356

66038

66801

65655

65606

41

69688

69140

68619

68140

67722

67379

67124

66966

66913

42

71182

70594

70034

69520

69070

68701

68426

68257

68200

43

72665

72036

71436

70884

70401

70005

69710

69527

69466

44

74137

73465

72822

72232

71715

71289

70972

70777

70711

45

74

ELLIPTIC FUNCTIONS

+

*=o

* = 5<>

= 10

* = 15

* = 20

= 25

= 30

= 35

* = 40

= 45<>

46

80285

80228

80056

79775

79390

78911

78350

77721

77040

76326

47

82030

81969

81787

81489

81081

80573

79977

79308

78584

77824

48

83776

83711

83518

83202

82770

82231

81599

80890

80121

79313

49

85521

85453

85249

84914

84457

83887

83217

82466

81651

80794

50

87266

87194

86979

86626

86142

85539

84832

84036

83173

82265

51

89012

88936

88709

88336

87826

87189

86442

85601

84689

83728

52

90757

90677

90438

90045

89507

88836

88048

87161

86197

85182

53

92502

92418

92166

91753

91187

90481

89650

88715

-87698

86627

64

94248

94159

93895

93460

92865

92122

91248

90264

89193

88063

55

95993

95900

95622

95166

94541

93761

92843

91807

90680

89490

56

97738

97641

97350

96872

96216

95397

94433

93345

92160

90908

57

99484

99381

99077

98576

97889

97030

96019

94878

93634

92318

58

1-01220

1-01122

1 '00803

1-00279

99560

9H061

97602

96405

95100

93719

59

1 '02974

1-02863

1-02529

1-01981

1 '01 229

1-00289

99180

97928

96560

95111

60

1-04720

1-04603

1-04255

1-03683

1'02897

1-01915

1-00756

99445

98013

96495

61

1*06465

1-06343

1-05980

1-05383

1-04563

1-03538

1-02327

1-00957

99460

97871

62

1-08210

1-08084

1-07705

1-07083

1-06228

1-05J5H

1-03895

1-02165

1'00900

99238

68

1-09956

1-01)824

1-09430

1-08781

1-07H91

1-06776

1-05459

1-03967

1-02334

1-00598

64

1-11701

1-11564

1-11154

1-10479

1-09553

1-08392

1-07020

1-05465

1-03762

1-01949

65

1-13446

1-13304

1-12878

1-12176

I '11213

1-10005

1-08577

1-06958

1-05183

1-03293

66

1-15192

15013

14601

1-13873

12871

1-11616

1-10132

1-08447

1-06599

1-04629

67

I'lliiW

16783

16324

1-15568

14529

1-13225

1-11683

1-09932

1-08009

1-05957

68

1- 18(18-2

18,") 23

18047

1-17263

16186

1-14832

1-1.-5231

1-11412

1-09143

1-07279

69

1-20428

20202

19769

1-18957

17839

1-1 (.437

1-14776

1-12888

1-10812

1-08593

70

1-22173

22002

1-21491

1-20650

19493

1-18040

1-16318

1-14360

1-12205

1-09901

71

1-23918

23741

23213

1-22343

21145

1-19640

1-17857

1-15828

1-13594

1-11202

72

1-25664

25481

24935

1-24034

22796

1-21239

1-19391

1-17293

1-14977

1-12497

73

1-27409

27220

20656

1-25726

24416

1-22837

1-20928

1-18754

1-16356

1-13786

74

1-29154

28959

28377

1-27417

2(5094

1-24132

1-22459

1-20211

1-17731

1-15068

75

1-30900

1-30698

30097

1-29107

27742

1-26026

1-23989

1-21666

1-19101

M6346

78

1-32645

1-32437

1-31818

1-30796

29389

1-27619

1-25516

1-23117

1-20467

M7618

77

1-34390

1-34176

1-33538

1-32486

31035

1-29210

1-27041

1-24566

1-21830

1-18885

78

1-36136

1-35915

1-35258

1-34174

32G80

1-30800

1-28565

1-26012

1-23189

1-20148

79

1-37881

1-37654

1-36978

1-35862

34325

1-32389

1-30086

1-27456

1-24544

1-21407

80

1-39626

1-39393

1-38698

1-37550

35968

1-33976

1-31606

1-28897

1-25897

1-22661

81

1-41372

1-41132

1-40417

1-39238

1-37611

1-35563

1-33124

1-30336

1-27246

1-23912

82

1-43117

1-42871

1-42137

1-40925

1-39254

1-37148

1-34641

1-31773

1-28594

1-25159

83

1-44862

1-44610

1-43856

1-42612

1-40896

38733

1-36157

1-33209

1-29939

1-26404

84

1-46608

1-46349

1-45575

1-44299

1-42537

40317

1-37672

1-34643

1-31282

1-27646

85

1-48353

1-48087

1-47294

1-45985

1-44178

1-41900

1-39186

1-36076

1-32623

1-28886

86

1-50098

1-49826

1-49013

1-47671

1-45819

43483

1-40699

1-37508

1-3&63

1-30124

87

1-51844

1-51565

1-50732

1-49357

1-47459

1-45066

1-42211

1-38939

1-35302

1-31360

88

1-53589

1-53304

1-52451

1-51043

1-49100

1-46648

1-43723

1-40369

1-36640

1-32596

89

1-55334

1-55042

1-54170

1-52729

1-50740

1-48230

1-45235

1-41799

1-37977

1-33830

90

1-57080

1-56781

1-55889

1*54415

1-52380

1'49811

1-46746

1-43229

1-39314

1-35064

ft continued.

75

= 50

= 650

= 60

= 65

= 70

= 75

= 80

= 85

= 00

*

75599

74881

74195

73564

73010

72554

72215

72005

71934

46

77050

76285

75553

74879

74287

73800

73436

73211

73135

47

78490

77676

76896

76177

75546

75025

74636

74396

74314

48

79920

79054

78225

77459

76786

76230

75815

75558

75471

49

81338

80419

79538

78724

78007

77414

76971

76697

7U604

50

82746

81772

80836

79971

79208

78578

78106

77814

-.7715

51

84143

83111

82120

81202

80391

79720

79218

78907

78801

52

85529

84438

83388

82415

81554

80842

80307

79976

79864

53

8G904

85752

84641

83610

82698

81941

81374

81021

80902

54

88269

87052

85879

84788

83822

83020

82417

82042

81915

55

89622

88340

87101

85949

84926

84076

83436

83039

82904

56

90965

89614

88308

87092

86011

85110

84432

84010

838(57

57

92297

90876

89500

88217

87075

86122

85404

84957

84805

58

93619

92125

90677

89:525

88119

87112

86352

85878

85717

59

94930

93362

91839

90415

89144

88080

87276

86773

86603

60

96231

9458G

92986

91488

90148

89025

88175

87643

874G2

61

97521

05797

94118

92543

91132

89948

89049

88486

88 k 295

62

98802

96996

95236

93581

92096

90848

89898

89303

89101

63

1-00072

98183

96339

94602

93041

91725

90723

90094

89871)

64

1-01333

91)358

97427

-95606

93965

92580

91523

90858

90631

65

1-02585

1 '00522

98502

96593

94870

93412

92297

91595

91355

66

1-03827

1-01674

99562

97564

95756

94222

9:5047

92305

92050

67

1-05060

1-02815

1-00609

1)8518

96G22

95010

93771

92987

92718

68

1-06284

1 03945

1-01643

99456

97469

95775

94470

93642

9:5358

69

1-07500

1-05064

1-02664

1-00379

98298

96519

95144

94270

93969

70

1-08707

1-06173

1-03672

1-01286

99108

97240

95793

94870

91552

71

1-09907

1-07272

1-04668

1-02178

99900

97940

90417

95442

9510G

72

1-11098

1-08362

1-05651

1-03056

1-00674

98619

97016

95987

95G30

73

12283

1-09442

1-06624

1-03919

1-01431

99278

97590

96503

9G12G

74

13460

1-10513

1-07586

1-04709

1-02172

99916

98141

96992

9G593

75

14631

1-11577

1-08537

1-05607

1-02896

1-00534

98667

97453

97030

76

15795

1-126,'{2

1-09478

1-06432

1-03605

1-01133

99170

97887

97437

77

16954

1-13680

1-10410

1-07245

1-04300

1-01714

99650

98293

97815

78

18107

1-14721

1-11333

1-08047

1-04981

1-02277

1-00107

98671

981G3

79

19255

1-15755

1-12249

1-08839

1-05648

1-02823

1-00543

99023

98481

80

1-20399

1-16784

1-13156

1-09621

1-06304

1-03354

1-00958

99318

98769

81

1-21538

1-17807

1-14057

1-10395

1-06948

1-03870

1-01354

99646

99027

82

1-22673

1-18825

1-14952

1-11161

1-07582

1-04372

1-01731

99920

99255

83

1-23805

1-19839

1-15841

1-11920

1-08207

1-04863

1-02091

1-00168

99452

84

1-24934

1-20850

1-16726

1-12673

1-08825

1-05343

1-02436

1-00394

99619

85

1-26061

l-2i857

1-17606

13421

1-09135

1-05813

1-02768

1-00598

99756

86

1-27186

1-22862

1-18484

14165

1-10041

1-06277

1-03089

1-00784

99863

87

1-28310

1-23865

1-19359

1490G

1-10642

1-06735

1-03401

1-00954

99939

88

1-29432

1-24867

1-20233

15645

1-11241

1-07188

1-03708

1-01113

99985

89

1-30554

1-25868

1-21106

16383

1-11838

1-07641

1-04011

1-01266

1-00000

90

7 6

XXVI. COMPLETE ELLIPTIC FUNCTIONS.

e

F, .

E,

F,

E,

e

Fi

Ei

GO

1-57080

1-57080

30

1-68575

1-46746

60

2-15652

1-21106

1

1-57092

1-57068

31

69411

1-46077

61

2-18421

1-20154

2

1-57127

1-57032

32

70284

1-45391

62

2-21319

1-19205

3

1-57187

1-56972

33

71192

1-44687

63

2-24355

1-18259

4

1-57271

1-56888

34

72139

1-43966

64

2-27538

1-17318

6

1-57379

1-56781

35

73125

1-43229

65

2-30879

1-16383

6

1-67511

1-56650

36

74150

1-42476

66

2-34390

1-15455

7

1-57668

1-56495

37

1-75217

1-41707

67

2-38087

1-14535

8

1-67849

1-56316

38

1-76326

1-40924

68

2-41984

1-13624

1-58054

1-56114

39

1-77479

1-40126

69

2-46100

1-12726

10

1-58284

1-55889

40

1 -78(577

1-39314

70

2-50455

1-11838

11

1-58539

1-55640

41

1-79922

1-38189

71

2-55073

1-10964

12

1-68820

1-55368

42

1-81216

1-37650

72

2-.V.W82

1-10106

13

1-59125

1-55073

43

1-82560

1-36800

73

2-65214

1-09265

14

1-69457

1-54755

44

1-83957

1-35938

74

2-70807

1-08443

15

1-59814

1-54415

45

1-85407

1-35064

75

2-76806

1-07641

10

1-60198

1-51052

46

1-86915

1-34181

76

2-8.5267

1-06861

17

1 -60008

1-53667

47

1-88481

1-33287

77

2-90256

1-06106

18

1-61045

1-532(50

48

1-90108

1-32384

78

2-97857

1-05378

19

1-61610

1-52831

49

1-91800

1-31473

79

3-06173

1-04679

20

1-62003

1-52380

50

1-93558

1-30554

80

3-15339

1-04011

21

1-62523

1-51908

51

1-95386

1-29628

81

3-25530

1-03379

22

1-63073

1-51415

52

1-97288

1-28695

82

3-36987

1-02784

23

1-63652

1-50901

53

1-99267

1-27757

83

3-50042

1-02231

24

1-64260

1-50366

64

2-01327

1-26815

84

3-651,86

1-01724

25

1-64900

1-49811

55

2-03472

1-25868

85

3-83174

1-01266

26

1-65670

1-49237

56

2-05706

1-24918

86

4-05276

1-00865

27

1-6G272

1-48643

57

2-08036

1-23966

87

4-33865

1-00526

28

1-67006

1-48029

58

2-10466

1-23013

88

4-74271

1-00258

29

1-67773

1-47397

59

2-13002

1-22059

89

5-43491

1-00075

90

Inf.

1-00000

77

xxvii Loo r(/).

X

1

2

3

4

5

6

7

8

ro

T-9( )

9753

9513

9280

9053

8834

8621

8415

8215

8021

1-1

I- 978:34

7653

7478

7310

7147

6990

6839

6694

6551

6121

1-2

T-96292

6169

6052

5910

5833

5732

5636

5545

6459

5378

1-3

1 '95302

5231

5165

5104

5047

4995

4948

4905

4868

4834

T4

1-94805

4781

4761

4745

4734

4726

4724

4725

4731

4741

1-5

1-94754

4772

4791

4820

4850

4884

4921

4963

5008

5057

re

1-95110

5167

5227

5'29 1

5359

5430

5505

5583

5665

5700

1'7

1-95839

6931

6027

6126

6229

6335

6444

6556

6672

6791

1-8

1-96913

7038

7167

7298

7433

7571

7712

7856

8004

8154

T9

1-98307

8463

8622

8784

89-10

9117

9288

9162

9(138

9818

XXVIII. Loo I (x \ i).

X

Lou i (x + 1)

x

Lou r(.c+ 1)

x

Lou i'U + 1)

.r

Loo I(JH 1)

I

-00000

26

26*60502

51

66-190(55

76

111-27513

2

30103

27

28'0,'i698

52

67-90665

77

11 3- 16! 92

3

77815

28

29-4X414

53

69-03092

78

115-05401

4

1-38021

29

30-94654

54

71-36332

79

116-95164

5

2-07918

30

32-42366

55

73-10368

80

118-85473

6

2-85733

31

33-91502

56

74-85187

81

120-76321

7

3-70213

32

35-42017

57

76-60774

82

122-67703

8

4-60552

33

36 -93809

58

78-37117

83

124-59610

9

5-55976

34

38' 17016

59

80-14202

84

126-52038

10

6*55976

35

40-01423

60

81-92017

85

128-44980

11

7-60116

36

41-57054

61

83-70550

86

130-38430

12

8-68034

37

43-13874

62

85-49790

87

132-32382

13

9-79128

38

44-71852

63

87-29721

88

134-26830

14

10-94041

39

46-30959

64

89-10842

89

136-21769

15

12-11650

40

47-91165

65

90-91 633

90

138-17194

16

13-32002

41

49-52443

66

92-735*7

91

140-13098

17

14-55107

42

51-14768

67

94-56195

92

142-09477

18

15-80634

43

52-78115

68

96-39446

93

144-06325

19

17-08509

44

54-42460

69

98-23331

94

146-03638

20

18-38612

45

56-07781

70

100-07841

95

148-01410

21

19-70834

46

57-74057

71

101-02966

96

149-09637

22

21-05077

47

59-41267

72

103-78700

97

151-98314

23

22-41249

48

61-09391

73

105-06032

98

153-97437

24

23-79271

49

62-78410

74

107-51956

99

155-97000

25

25-19065

50

64-48307

75

109-39461

100

157-97000

XXIX. ZONAL SURFACE HARMONICS.

X

PiGO

P(*)

Pa(*)

P(*)

P.(*)

P.() '

Pi()

00

+00

-50000

-00000

+'37500

+00000

-31250

+'00000

01

+01

-49985

-01500

+37463

+01874

-31184

-02186

02

+02

-49940

- -02998

+37350

+03743

-30988

- -04359

03

+03

- -49865

-04493

+37163

+05601

-30661

- -06509

04

+04

-49760

-05984

+36901

+07444

- -30205

- '08624

05

+05

-49625

-07469

+36565

+09266

-29622

- 10693

06

+ 00

- '49460

-08946

+36156

+11062

-28913

- -12703

07

+07

- -49205

-10414

+35673

+12826

-28081

- -14644

08

+08

- -49040

-11872

+35118

+14555

-27130

- 16506

09

+09

-48785

-13318

+34491

+16242

- -26063

- 18278

10

+10

-48500

-14750

+33794

+17883

- -24883

-19949

11

+11

-481X5

-16167

+ -33027

+19473

- -23595

-21511

12

+12

- -47840

- -17568

+ 32191

+21008

-22204

-22955

13

+13

-47465

-18051

+ 31287

+22482

-20715

-24271

14

+14

-47060

-203 14

+30318

+23891

-19133

-25453

15

+15

- '46025

-2 1056

+29284

+25232

-17465

-26492

16

+16

-46160

-22976

+ '28187

+26499

-15716

-27383

17

+17

- '45665

-21272

+27028

+27688

-13894

-28119

18

+18

-45140

-25542

+25809

+28796

-12005

-28695

19

+19

-44585

- -20785

+24533

+29818

-10057

-29107

20

+20

-44000

-28000

+23200

+30752

-08058

-29352

21

+21

- -43385

-29185

+21813

+31593

-00014

-29426

22

+22

- -42740

- -30338

+20375

+32339

- -03936

-29327

23

+23

- -42065

-31458

+18887

+32986

-01830

-29055

24

+24

-41360

-32544

+17352

+33531

+00294

-28610

25

+25

- -40625

-33594

+15771

+33972

+02428

-27992

26

+20

- -39860

- -34606

+14149

+34307

+01562

-27203

27

+27

- -39065

- -35579

+12488

+34532

+06687

-26246

28

+28

- -382-10

-36512

+10789

+34647

+08795

-25124

29

+29

-37385

-37403

+09057

+34650

+'10875

-23843

30

+30

- -36500

- -38250

+07294

+34539

+12918

-22407

31

+31

- '35585

- -39052

+05503

+34312

+14915

-20824

32

+32

- -34640

- -39808

+03688

+ 33970

+16856

-19100

33

+33

- -33605

-40516

+01851

+ 33512

+18732

-17244

34

+34

- -321*60

-41174

- -00004

+32937

+20534

-15266

35

+35

-31625

-41781

- -01872

+32245

+22251

-13176

36

+36

-30500

-42336

- -03752

+31438

+23875

-10984

37

+37

-29105

- '42837

- -05638

+30514

+25397

-08704

38

+38

-28340

- -43282

- -07528

+29477

+26808

-06347

39

+39

-27185

- -43670

-09416

+28326

+28100

-03927

40

+40

- -26000

- -44000

-11300

+27064

+29264

-01459

41

-r-41

-24785

-4 4270

-13175

+25693

+30291

+01042

42

+42

-23540

-44 178

-15036

+24215

+31176

+03561

43

+43

-22265

- -44623

-16880

+22633

+31909

+06082

44

+44

-20960

- -44704

-18702

+20951

+32486

+08587

45

+45

-19625

-44719

-20497

+19172

+32898

+11060

46

+46

-18260

-44666

-22261

+17301

+33141

+13483

47

+47

-16865

- -44544

-23989

+15341

+33209

+15838

48

+48

-15440

-44352

-25676

+13298

+33098

+18107

49

+49

-13985

-44088

-27316

+11177

+32804

+20272

60

+50

-12500

-43760

-28906

+08984

+32324

+22314

79

ZONAL SURFACE

PiM

PM

Pi(*)

PW

PiW

P.(*)

Pf()

51

+51

-10985

-43337

-30440

+06726

+81665

+24217

52

+'52

-09440

-42848

-31912

+04409

+30796

+25961

53

+53

-07865

-42281

-33317

+02041

+29747

+27530

54

+54

-06260

-41634

-34649

-00372

+28506

+28906

55

+56

' '04625

-40906

-35904

-02819

+27077

+30074

56

+56

-02960

-40096

-37074

-05294

+25460

+31016

57

+57

-01265

-39202

- -38155

-07786

+23660

+31719

58

+58

+00460

- -38222

-39140

-10285

+-216S1

+32169

59

+59

+02215

-37155

- '40024

-12781

+19528

+32353

60

+-GO

+04000

-36000

-40800

-15264

+17210

+32260

61

+61

+05815

-34755

-41462

-17721

+14733

+31880

62

+62

+07660

-33118

-42004

-20142

+12109

+31207

63

+63

+09535

-31988

-42418

-'22512

+09348

+30232

64

+64

+11440

-30164

-42700

-24819

+00402

+28954

65

+65

+13375

-28844

-42841

-27019

+03467

+27371

66

+66

+15340

-27126

-42836

-29188

+00379

+25483

67

+67

+17335

-25309

-42076

-'31220

-02785

+23295

68

+68

+19300

-23392

-42356

-'33131

-00006

+20813

69

+ 69

+21415

-21373

-41809

-34903

-09201

+18049

70

+70

+23500

-19250

-41206

- -30520

-12529

+15016

71

+'7i

+ 25615

-17022

-40361

-37964

-15782

+11731

72

+72

+ 27760

-14088

-39327

-39217

-18901

+08217

73

+'73

+29935

-12246

-38095

-4021,0

- -22130

+04499

74

+74

+ 32140

- -09694

-30059

-4 1074

-25175

+00609

'75

+75

+ 34375

-07031

- '35010

-41638

-28078

-03418

76

+76

+36640

-04256

-33140

-41931

-30807

-07541

77

+77

+38935

-01307

-31043

-41932

-33325

-11713

78

+78

+41260

+01638

-28709

-41018

-3f>r>H9

-15881

79

+79

+ 43615

+04760

-26131

-4()%G

-37557

-19987

80

+80

+46000

+'08000

-23300

-39952

-39180

-23965

81

+81

+48415

+ 11360

-20208

-38552

-40409

-27743

82

+82

+50860

+ 14842

-10847

- -36739

-41193

-31240

83

+83

+ 53335

+ 18447

-13207

-34489

-4 1475

-34368

84

+84

+55840

+22176

-09281

-31774

-41198

-37033

85

+85

+ 58375

+26031

- -050t>0

- -28506

-10300

-39130

86

+86

+60940

+30014

-00534

-24838

-38716

-40545

87

+87

+ 63535

+34126

+01305

-20559

-30379

-41150

88

+88

+ 66160

+38308

+09467

-15099

-33217

-40829

*89

+'89

+68815

+42742

+14960

-10228

-29156

-39423

90

+90

+'71500

+47250

+20794

-04114

-24116

-36782

91

+91

+74215

+51893

+'26978

+02076

-18018

- -32743

92

+92

+70960

+56672

+33522

+10175

-10774

-27121)

93

+93

+'79735

+'61589

+'40435

+18417

-02295

-19749

94

+94

+ 82540

+66646

+47728

+27438

+07512

-10404

'95

+95

+85375

+71844

+55409

+37274

+18745

+01123

96

f 96

+88240

+77184

+63489

+47962

+31506

+15000

97

+97

+91135

+82668

+71978

+59539

+45899

+31650

'98

+98

+94060

+88298

+80886

+72045

+62035

+51151

99

+99

+97015

+94075

+90223

+85518

+80029

+73838

TOO

+1-00

+1-00000

+1-00000

+1-00000

+1-00000

+1-00000

+ 1 '00000

So

XXX. BESSEL FUNCTIONS.

X

Jt(*)

JiW

X

JoCO

JiC*)

X

Jt(*)

Ji(*>

O'O

+ 1-00000

+00000

5-0

-17760

-32758

10-0

-24594

+04347

O'l

-f- -99750

+04994

5*1

-14433

-33710

10-1

-24903

+01840

0*2

+99002

+09950

6'2

-11029

-34322

10'2

-24962

-00662

0'3

+977(53

+14832

5'3

-07580

-34596

10-3

-'24772

-03132

0'4

+96040

+19603

5'4

-04 121

-34534

10-4

-24337

-05547

0-6

+93847

+24227

5*5

-00684

-34144

10'5

-23665

-07885

0-6

+91200

+28670

5'6

+02697

-33433

10'6

-22764

-10123

0'7

+-H8120

+32900

6'7

+'05992

-32415

10-7

-21644

-12240

0'8

+84629

+30884

5'8

+ 09170

-31103

10-8

-20320

-14217

0-9

+80752

+40595

5'9

+11203

-29514

10'9

- 18806

-16035

ro

+76520

+4400.)

6-0

+15065

-27668

11-0

-17119

-17679

I'l

+71902

+47000

6-1

+ 17729

-25586

ll'l

-15277

-19133

ra

+67113

+49829

6'2

+ 20175

-23292

11*2

-13299

- -20385

T3

+ 62009

+52202

6'3

+ 22381

-20809

11-3

-11207

-.21426

T4

+ 56686

+51195

6'4

+ 24331

-18164

11-4

-09021

-22245

1-5

+51183

+55794

6'5

+ 26009

- 15384

11-5

-06765

-22838

1-6

+455-10

+56990

6'6

+ 27404

-12498

11-6

-04462

-23200

1-7

+39798

+57777

6'7

+28506

- -09534

11-7

-02133

-23330

1-8

+331)1)9

+58152

6'8

+ '29310

-06522

11 '8

+ 00197

-23228

1-9

+28182

+ 58116

6'9

+ 29810

-03490

11-9

+02505

-22898

2'0

+22389

+57672

7'0

+ 30008

-00168

12-0

+04769

-22345

2-1

+16661

+ 56829

7-1

+ 29905

+02515

12-1

+ 06967

-21575

2'2

+'11036

+ 55596

7'2

+ 29507

+05433

12'2

+09077

-20598

2'3

+05554

+ 53987

7'3

+ 28822

+08257

12'3

+11080

-19426

2'4

+00251

+52019

7'4

+ 27860

+10963

12'4

+12956

-18071

2'5

-04838

+ 4 9709

7'5

+ 26634

+13525

12-5

+14688

-16548

2'6

-09680

+ 47082

7'6

+ 25160

+15921

12-6

+16261

-14874

2'7

-14245

+ 44160

7'7

+ 23456

+18131

12'7

+17659

-13066

2'8

-18504

+ 40971

7'8

+ 21541

+20136

12'8

+18870

-11143

2'9

-22431

+37513

7'9

+ 19436

+21918

12'9

+19884

-09125

3'0

-26005

+33906

8'0

+17165

+ 23464

13-0

+20693

-07032

3'1

-29206

+ -30002

8-1

+14752

+24761

13-1

+21289

-04885

3*2

-32019

+ 26134

8'2

+ 12222

+25800

13-2

+21669

-02707

3'3

-34430

+ 22066

8'3

+ 09601

+26,074

13'3

+21830

-00518

3'4

-36430

+17923

8'4

+ 06916

+27079

13'4

+21773

+ 01660

3*5

-38013

+13738

8'5

+04194

+27312

13'5

+21499

+03805

3'6

-39177

+09547

8-6

+ 01462

+27275

13'6

+21013

+05896

3'7

-39923

+ 05383

8'7

-01252

+26972

13'7

+20322

+07914

3'8

-40256

+ 01282

8'8

03923

+26407

13-8

+19434

+09839

3'9

-40183

-02724

8'9

-06525

+25590

13'9

+18358

+11652

4'0

-39715

-06604

9'0

-09033

+24531

14*0

+17107

+ 13338

4'1

- -38867

-10327

9*1

-11424

+23243

14'1

+15695

+14878

4*2

- -37656

-13865

9*2

-13675

+21741

14'2

+14137

+16261

4'3

-36101

-17190

9'3

-15766

+20041

14'3

+12449

+17473

4'4

-34226

-20278

9'4

-17677

+18163

14-4

+10648

+18503

4*5

-'32054

-23106

9'5

-19393

+16126

14'5

+08754

+19343

46

-29614

-25655

9-6

-20898

+13952

14'6

+06786

+19985

4'7

-26!)33

-27908

9'7

-22180

+ 11664

14'7

+04764

+20425

4*8

- -24043

-298r>0

9'8

-23228

+ 09284

14-8

+02708

+ 20660

4-9

-20974

-31169

9*9

- -24034

+ 06837

14'9

+00639

+20688

6-0

- 17760

-32758

10-0

-24594

+04347

15-0

-01422

+ 20510

c>

I

o

125

t

fc\$;S
isites

IM i

i\$

+ + ++++4- 4-4-+++

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O r-t UO O OO C^

co oo oo io -< o5

O -H t O I H

++++"l "l

-
i" o t >-*t>.i>.a>t>.

+4-+l-4- l l l l 4-

8SS35
4-4-Vl "l *l

SIQT^COO -^ o> o h -^ <c>o>t--e^oo

o-^co tD^o-^boo cocot^coiN

OO^ C^ CO CO i I O C^C^O-HC^

4-++4- +4-4-4- "l "l "l *l ++

l + +

O O CO CO CO rj< O *G t> O "* O 00 ^
OOrH C^OOOCO*^ 1 tOO^OC^C^i
-TC^i-t r-il-t-^'O'O OSOfiMOS1O

QO ^ -^ -^ O t
rH tO I O O CD
OSC^OCiO O

O iO C?l O OOt-'O ig

(NOi O

>COf Oi

O * t C

+ + + + ++!

! l l +++

4 ' i CS *! *t'OT't>C s < T^COQOt"-^ O

i O "- -^ i^COQO'-tuO -^ 00 O Q D

-CO-<TH C^i-tOC^rH Oi-Or-< TH

" * " " * ' '

Tf O CO O CO O CO 5Ol--l'.iO

CO-^" ^t^C^I-CO OOiOSOOO

XNM pr-ic^o^H < ^ l< r >o 7*'r (
" " " * ' "

+++ I l l ++

l "l 4- ++! 'l "l +

oosoio oooo*oco t*a>o>cor^ ooooco <N

0*000^ tOOO^OCO OJ-H?OOSO C^CiQOCO^D O

^cooi> t^oo~<o o t^ to -^ ^TOCOO t--

^O^^O* t*OOt>.O> '*1'l>^'Ot s -Ht>O-'J* O

++'l "l 'l 4-++1 *l "l +++ "l 'l Vl + +

rHiHiHrHrH

>25 1

388

8S

> o o <:

O O *-" <N CO to

^ _ -

OOO OOOOO O rH <?> CO LO t^-
O O O OOOOO OOOOO O

I

?88

-^?l tO 5

"<* t" rH OO t- -( 00 t- <M tO

- O O r- 1 rH ^J H^!O30C'1OO "O

50 OOOOO O O O rH rH <M

o o o
O 00

>S.:2 g^2^- ^^^ot

OSOO ^ oooo^
OOOOO

w

PQ

I

.-CyiC-l^^ COI>.tOC5C?> rHtOrHCi-^ C^rH<MQ

5J <0 >0 O to OOCOril-tO C^CMtOOOO ?-l>.^C

hOO COr^OOt>-QO -H>
COO t>-CO*efCOO l^ t

>t*ooo r>-tocicio> -^cot>cooo ^ioor-ito coo^voco TH

SSt^ZJ"^ <MC5Trcjco (Mt^r-oio orfi-ntoco co-^Cicoo co

^^^OrH <Ncotocv^ rJSP.os:S9 ^999 co(N<Ntp<^ ^

THrHC^C^ COi^OtOCIO rH

5<Ml>t>O iCOOt-M*

?O(Mcoco i>.'-cioaip

5OOOO -rH(>J<MCO*b

?Zi2r% cSSS^S J

SO CO O <O 00 C5 OO CN tp^rH^.H ^ CO rH t* -^ vo to to CO CO tO

>OOt- OO rHtOOTfit^. O^OOlOrH CO'^'OOIO OSOftOiO CO

oo'rco<N *o^tooo*H cirHgsioo oeor>-C5Tt* iftt>.ooo<M co

S5S^2 55^959^ 4701^ t-co a>t^<pt^rH ^-^^^^ ^,

co co -^ o to oo

" rH -Tf O O
rH HrH (M

-O> <N<MOCO-1 O

4D O"t"rHCsOS rf"

JrH CO^OOt^. CM

OOOOO

OW^fCDOO OtN^QDOO O

CO CO COCO CO

XXXIII. ROOTS OF J n ( = 0.

n =

n = 1

n = 2

n = 3

n = 4

n = 5

1

2-405

3-832

5*135

6-379

7-586

8-780

2

5-520

7-016

8-417

9-760

11-064

12*339

3

8-654

10-173

11-620

13-017

14-373

15-700

4

11-792

13-323

14*796

16*224

17-616

18-982

5

14-931

16-470

17'960

19*410

20-827

22-220

6

18-071

19-616

21*117

22-583

21-018

25-431

7

21-212

22-760

24-270

25-749

27-200

28-628

8

24-353

25-903

27-421

28-909

30-371

31-813

9

27-494

29-047

30-571

32-050

33-512

34-983

ROOTS OF J (V) = 0, AND

= 0.

- FOR J (^ v ) =

- roil .1 )(.*') =

L i<ou J (.0 =

- FOR Ji() =

1

0-7655

1*2179

7

6-7519

7*2448

2

1-7571

2*2330

8

7-7516

8*2454

3

2-7546

3-2383

9

8-7514

9*2459

4

3-7534

4*2411

10

9-7513

10-2463

5

4-7527

5*2428

11

10*7512

11-2466

6

5-7522

6*2439

12

11-7511

12-2469

1

3
5

7
9

11
-1
-3

-5

-7

-9

-11

XXXIV.-.T i+i (r).

em x

sin x

- x cos x

/3 __\ ina; _3
/15^6\ . /1,5 \

f 10 ^ _ 1? i i\ _ (I 05 _ i Qx \
/045 420 15\ . /945 1

COS X

COS X

Bin

/15_6'
<105 45

/15

420 ,

XXXV.I = -?- I e-di

X

I

X

I

X

I

X

I

X

I

X

I

01

01128

51

52924

1-01

84681

1'51

96728

2-01

99552

2'51

99961

02

02256

52

53790

1'02

85084

1-52

96841

2-02

99572

2*52

99963

03

03384

53

54646

1'03

85478

1-63

96952

2-03

99591

2'53

99965

04

04511

54

55494

ro4

85865

1-54

97059

2'04

99609

2'54

99967

05

05637

55

66332

1'05

86244

1-55

97162

2-05

99626

2*55

99969

'06

06762

56

57162

1-06

86614

1'56

97263

2-06

99642

2'56

99971

07

07886

57

57982

1'07

86977

1'57

97360

2-07

99658

2'57

99972

08

09008

58

68792

1-08

87333

1'58

97455

2-08

99673

2'58

99974

09

10128

'59

59594

1'09

87680

1'59

97546

2'09

99688

2*59

99975

10

11246

60

60386

no

88021

1'60

97635

2'10

99702

2*60

99976

11

12362

61

61168

I'll

88353

rei

97721

211

99715

2'61

99978

12

13476

62

61941

ri2

88671)

1-62

97804

212

99728

2'62

99979

13

14587

63

62705

ri3

88907

1*63

97884

213

99741

2'63

99980

14

15695

64

63459

ri4

89.108

1-64

979(52

214

99753

2'64

99981

15

1G800

65

64203

115

896 12

T65

98038

215

99764

2'65

99982

16

17901

'66

64938

1-16

89910

T66

98110

216

99775

2'66

99983

17

18909

67

65663

1'17

90200

1'67

98181

217

99785

2'67

99984

18

20093

'68

66:578

1-18

90484

T68

98249

218

9979:.

2'68

99985

19

211H4

'69

67084

1-19

907(51

1-69

98315

219

99805

2*69

99986

20

22270

70

67780

1-20

91031

1'70

98379

2-20

99814

2*70

99987

21

23352

*71

68467

1-21

91296

1-71

98141

2'21

99822

2'71

99987

'22

24430

72

69143

T22

91553

T72

98500

222

90831

2'72

99988

*23

25502

73

69810

1-23

91805

1-73

98558

2'23

99839

2'73

99989

24

26670

74

70468

1-24

92051

T74

98(513

2'24

90816

2*74

99989

25

27633

75

71116

1-25

92290

T75

98G67

2-25

99854

2*75

99990

*26

28690

76

71754

1-26

92524

1-76

98719

2'26

99861

2'76

99991

27

29742

77

72382

1-27

92751

1'77

98769

2'27

99867

2-77

99991

28

30788

78

73001

1-28

92973

1-78

98817

2-28

99874

2'78

99992

29

31828

79

73610

1*29

93190

1-79

98864

2'29

99880

2*79

99992

30

32863

80

74210

1-30

93401

1'80

98909

2'30

99886

2*80

99992

*31

33891

81

74800

1-31

93606

1'81

98952

2*31

99891

2'81

99993

33

34913

'82

75381

1'32

93807

1-82

98994

2'32

99897

2'82

99993

*33

35928

83

75952

1-33

94002

1'83

99035

2-33

99902

2'83

99994

'34

36936

84

76514

1-34

94191

1-84

99074

2'34

99906

2'84

99994

35

37938

85

77067

1'35

94376

1'85

99111

2'35

99911

2*85

99994

'36

38933

86

77610

1-36

94556

1-86

99147

2'36

99915

2*86

99995

'37

39921

87

78144

1-37

94731

1-87

99182

2*37

99920

2'87

99995

38

40901

88

78669

1-38

94902

1'88

99216

2'38

99924

2'88

99995

39

41874

89

79184

1-39

95067

1-89

99248

2-39

99928

2*89

99996

40

42839

90

79691

1-40

95229

1'90

99279

2-40

99931

2*90

99996

41

43797

01

80188

1-41

95385

1'91

99309

2'41

99935

2-91

99996

42

44747

92

80677

1-42

95538

T92

99338

2-42

99938

2*92

99996

*43

45689

93

81156

1-43

95686

1'93

99366

2'43

99941

2*93

99997

44

46623

94

81627

1-44

95830

1-94

99392

2'44

99944

2'94

99997

45

47548

95

82089

1-45

95970

1*95

99418

2*45

99947

2*95

99997

46

48466

96

82542

1-46

96105

1'96

99443

2*46

99950

2'96

99997

47

49375

97

82987

1-47

96237

1'97

99466

2-47

99952

2'97

99997

48

50276

98

83423

T48

96365

1-98

99489

2*48

99955

2-98

99997

49

61167

99

83851

1-49

96490

1-99

99511

2*49

99957

2-99

99998

50

52050

TOO

84270

1-50

96611

2'00

99532

2*50

99959

3-00

99998

00

1-00000

XXXVI. Ei(a:),

X

EiCr)

i(-*)

Ci(x)

8i(x)

X

EiCr)

Ei(-*)

C1(J)

BiCr)

oo

00

00

00

+ 00000

50

+45422

-5 r >977

-17778

+49811

01

-4-01793

-4-03793

-4-02798

+01000

51

+48703

-54782

-16046

+50269

02

-3-31471

-3-35471

-3-33491

+02000

52

+51953

-53622

-14355

+51225

03

-2-89912

-2-95912

-2-92957

+03000

53

+55173

-52495

-12707

+62180

04

-2-60126

-2-68126

-2-64206

+04000

54

+58365

61400

-11099

+53133

05

^2-.%789

-2-46790

-2-41914

+-OJ999

55

+6 1529

-50336

- -09530

+54084

06

-2-17528

-2-29531

-2-23709

+05999

56

+(14668

-49302

-07999

+65033

07

-2-01080

-2-15084

-2-08327

+06998

57

+67782

-482%

- -06504

+55981

08

-1-86688

-2-02694

-1-95011

+07997

58

+'70873

-47317

- -05044

+56927

09

-1-73866

-1-91874

-1-83275

+08996

59

+73941

-46365

- -03619

+'57871

10

-1-62281

-1-82292

-1-72787

+09994

60

+76988

-45438

- -02227

+ 58813

11

-1-51696

-1-73711

-1-63308

+10993

81

+80015

-44535

--OOH68

+59763

12

-1-41935

-1-65954

-1-54665

+11990

62

+83023

-13656

+ 004(51

+60691

13

-1-32866

-1-58890

-1-46723

+12988

63

+86012

-42800

+01768

+61627

14

-1-24384

-1-52415

-1-39379

+13985

64

+88984

-41965

+03026

+62661

15

-1-16409

-1-46446

-1-32552

+14981

65

+91939

-4 11 52

+ -04265

+63494

16

-1-08873

-1-40919

-1-26176

+ 15977

66

+94878

- -40359

+ 05476

+64424

17

-1-01723

-1-35778

-1-20196

+ 1(5973

67

+97802

-39585

+0(5659

+65351

18

-94915

-1-30980

1-14567

+ 17968

68

+ 1-00712

- 38831

+07816

+66277

19

-88410

-1-26486

-1-09253

+18962

69

+1-03608

-'38095

+08946

+67201

20

- -82176

-1-22265

-1-04221

+19956

70

+1-06491

-37377

+10051

+68 122

21

-76187

-1-18290

-99444

+20949

71

+ 1-09361

-36676

+11 132

+69041

22

-70420

-1-14538

-94899

+ 21941

72

+ 1-12220

-35992

+12188

+69968

23

-64853

-1-10988

- -90566

+22933

73

+ 1-15068

-35324

+13220

+70873

24

- -59470

-1-07624

-86427

+23923

74

+1-17906

-34671

+14230

+71786

25

-54254

-1-04428

-82466

+24913

75

+ 1-20733

- -34034

+ 15216

+72696

26

-49193

-1-01389

- -78671

+25903

76

+1-23551

-33412

+16181

+73603

27

-44274

-98493

- 75029

+26891

77

+ 1-2(5360

-32803

+17124

+74608

28

-39486

- 96731

-71529

+27878

78

+1-20161

-32209

+18046

+76411

29

-34820

- -93092

- -68161

+28865

79

+ 1-31954

-31628

+18947

+76312

30

- -30267

- -90568

-64917

+29850

80

+1-34740

-31060

+19828

+77210

31

- -25819

-88151

-61790

+ -308:15

81

+ 1-37518

-30504

+ 20689

+78105

32

- -21468

-85834

-68771

+31819

82

+ 1-40290

-29961

+21530

+78998

33

-17210

- 83610

-55855

+32801

83

+ 1'4305G

-29430

+22353

+79888

34

-13036

- -81475

- -53036

+33782

84

+1-45816

-28910

+23157

+80776

35

-08943

-79422

- 50308

+34763

85

+1-48571

- -28402

+23942

+81661

36

-04926

-77446

-47666

+35742

86

+ 1-51322

-27905

+24710

+82544

37

-00979

-75544

-45107

+36720

87

+ 1-54067

- -27418

+25460

+83424

38

+ '02901

-73711

-42625

+37696

88

+1-56809

- -26941

+26192

+84801

39

+06718

- -71944

-40218

+38672

89

+1-59547

-26475

+26908

+86175

40

+10477

- 70238

-37881

+39646

90

+1-62281

-26018

+27607

+86047

41

+14179

- -68591

-35611

+40619

91

+1-66013

-25671

+28289

+86916

42

+17828

-67000

-33406

+41591

92

+1-67741

-25134

+28956

+87782

43

+21427

-65461

-31262

+42561

93

+1-70468

- -24705

+29606

+88646

44

+24979

-63973

-29178

+43529

94

+1-73192

-24285

+30242

+89606

45

+28486

-62533

-27149

+44497

95

+1-75915

-23874

+30861

+90364

46

+31950

-61 139

- -25175

+45463

96

+1-78636

- -23471

+31466

+91219

47

+35374

-59788

-23253

+46427

97

+1-81356

-23076

+32056

+92070

48

+38759

-58478

-21380

+47390

98

+1-84075

-22689

+32632

+92919

49

+42108

-57209

-19566

+48351

99

+1-86793

-22310

+33193

+93765

50

+45422

-55977

-17778

+49311

1-00

+1-89512

-21938

+33740

+94608

86

X

Ei(x)

Ei(-s)

Ci(*)

Si(*)

X

Ci(*)

Si()

ro

+1-89512

-21938

+33740

+94608

20

+0444

+1-5482

1-1

+2-16738

- -18599

+38487

+1-02869

25

-0068

+1'5315

ra

+2-44209

--15841

+42046

+ 1-10805

30

-0330

+1-5668

ra

+2-72140

- 13545

+44574

+1-18396

35

-0115

+1-5969

1-4

+3-00721

11622

+46201

+ 1-25623

40

+ 0190

+1-5870

1*6

+3-30129

- -10002

+47036

+1*32468

45

+0186

+1-5587

re

+3-00532

- -08631

+47173

+1-38918

50

-0056

+1-5516

1-7

+3-92096

- -07465

+46697

+1*44959

55

-0182

+1-5707

1-8

+4-24987

- -06471

+ -45681

+ 1-50582

60

-0048

+1-5867

1-9

+4-59371

- -05620

+44194

+1-55778

65

+0128

+1-5775

2-0

+4-95423

- -04890

+42298

+1-60541

70

+0109

+1-5616

2-1

+5-33324

-04261

+40051

+1-64870

75

-0053

+1-5586

2*2

+5-73261

- -03719

+ -37507

+1-68762

80

-0124

+ 1-5723

2'3

+6-15438

- -03250

+34718

+1-72221

85

--0019

+ 1-5824

2'4

+6-60067

- -02844

+31729

+1-75249

90

+0100

+1-5757

2'5

+7-07377

-02491

+28587

+ 1-77852

95

+0071

+1-5630

2*6

+7-57611

-02 185

+25334

+ 1-80039

100

-0051

+1-5622

2'7

+8-1 1035

-01918

+ 22008

+1-81821

110

-0003

+1-5799

2'8

+8-67930

-01685

+ 18649

+ 1-83210

120

+ 0048

+1-5640

2'9

+9-28602

- -01482

+15290

+ 1-84219

130

0071

+1-5737

8'0

+9-93383

-01305

+11963

+1-84865

140

+0070

+1-5721

31

+10-6263

-01149

+ -08699

+1-85166

150

-0048

+1-5662

3*2

+11-3673

-01013

+ -05526

+ 1-85140

160

+0014

+1-5769

3*3

+12-1610

- -00894

+ -02468

+ 1-81808

170

+ 0020

+ 1-5653

3'4

+13-0121

-00789

-00 152

+1-84191

180

-0044

+1-5741

3'5

+13-9254

- -00697

- -03213

+ 1-83313

190

+0052

+ 1-5704

3'6

+14-9063

- -00616

- -05797

+ 1-82195

200

-0044

+1-5684

3*7

+15-9606

- -00545

-08 190

+ 1-80862

300

-0033

+1-5709

3*8

+17-0948

- -00482

- -10378

+1-79339

400

-0021

+1-5721

3-9

+18-3157

- -00427

- -12350

+ 1-77650

500

-0009

+1-5726

4-0

+19-6309

-00378

- -14098

+ 1-75820

600

+0001

+1-5725

41

+21-0485

- 00335

-15617

+ 1-73874

700

+0008

+1-5720

4'2

+22-5774

-00297

- -16901

+ 1-71837

800

+0011

+1-5714

4'3

+24-2274

- -00263

-17951

+ 1-69732

900

+0011

+1-5707

4'4

+26-0090

-00234

- -18766

+1-67583

1000

+0008

+1-5702

4'5

+27-9337

- -00207

- -19349

+ 1-65414

2000

+0005

+1-5710

4*6

+30-0141

-00181

-1970S

+ 1-63216

3000

+0001

+1-5711

4.7

+32-2639

-001 64

-19839

+1-61101

4000

-0002

+1-5710

4-8

+34-6979

-001 45

-19760

+1-58998

5000

-0002

+1-5708

4'9

+37-3325

-001 29

- -19478

+ 1-56956

6000

-0001

+1-5706

5'0

+40-1853

-00115

-19003

+1-54993

7000

+0001

+1-5707

6

+85-990

-00036

-06806

+ 1-42469

8000

+0001

+1-5708

7

+ 191-50

-00012

+07670

+1-45460

9000

+0001

+1-5709

8

+440-38

-00004

+12243

+1-57419

10,000

-0000

+1-5709

9

+1037-9

-00001

+05535

+1-66504

100,000

+0000

+ 1-5708

10

+2492-2

-00000

-04546

+1-65835

1,000,000

-oooo

+ 1-5708

11

+6071-4

-00000

-08956

+1-57831

12

+ 14960

00000

-04978

+1-50497

13

+37198

00000

+02676

+1'49936

14

+93193

00000

+06940

+1-55621

15

+234956

00000

+04623

+1-61819

XXXVIL-BINOMIAL COEFFICIENTS FOR INTERPOLATION
BY DIFFERENCES.

t

COEF
A 2

VS. OF
A3

t

COEF]

A 2

PS. OF
A 3

t

COKF
A 2

'3. OF

A 3

t

CORFl

A a

PS. OF
A 3

01

-005

+ 003

26

-096

+056

51

-125

+062

76

-091

+ 038

02

-010

-f-006

27

-099

+057

52

-125

+062

77

-089

+036

03

-015

+ 010

28

-101

+ 058

53

-125

+061

78

-086

+035

04

-019

+013

29

-103

+ 059

54

-121

+060

79

-083

+033

05

-024

+015

80

-105

+060

55

-124

+060

80

-080

+032

06

-028

+018

31

-107

+060

-56

-124

+059

81

-077

+031

07

-033

+021

32

-109

+061

57

-123

+058

82

-074

+029

08

-037

+024

33

-111

+062

58

-122

+058

83

-071

+028

09

-041

+026

34

-112

+ 062

59

-121

+057

84

-067

+026

10

-045

+028

35

-114

+063

60

-120

+056

85

-064

+024

11

-049

+031

36

-115

+063

61

-119

+055

86

-060

+023

12

-'053

+033

37

-117

+063

62

-118

+054

87

-057

+021

13

-057

+035

38

-118

+064

63

-117

+053

88

-053

+020

14

-060

+037

39

-119

+064

64

-115

+052

89

-019

+018

15

-064

+039

40

-120

+064

65

-114

+051

90

-045

+016

16

-067

+041

41

-121

+064

66

-112

+050

91

-041

+015

17

-071

+043

42

-122

+064

67

-111

+049

92

- -037

+013

18

-074

+045

43

-123

+064

68

-109

+048

93

-033

+012

19

-077

+046

44

-123

+064

69

-107

+047

94

-028

+010

20

-080

+ 048

45

-124

+064

70

-105

+045

95 '

-024

+008

21

-083

+049

46

-124

+064

71

-103

+044

96

-019

+007

22

-086

+051

47

-125

+064

72

-101

+043

97

-015

+ 005

23

-089

+052

48

-125

+063

73

-099

+042

98

-010

+003

24

-091

+053

49

-125

+063

74

-096

+040

'99

-005

+002

25

-094

+055

50

-125

+ 063

75

-094

+039

1-00

-000

+000

83

XXXVIII. - PROPORTIONAL PARTS.

X'l

X'2

x-3

x-4

x-5

x-6

X'7

x*8

X*9

10

1-0

2-0

3-0

4-0

5-0

6-0

7-0

8-0

9-0

11

M

2-2

3-3

4.4

5-5

6-6

7*7

8-8

9-9

12

1-2

2-4

3-6

4*8

6-0

7-2

8-4

9-6

10-8

13

1-3

2-6

3-9

6-2

G-5

7-8

9-1

10-4

11-7

14

1-4

2-8

4-2

5-6

7-0

8-4

9-8

11-2

12-6

15

1*5

3-0

4-5

6-0

7-5

9-0

10-5

12*0

13-5

16

1-6

3-2

4-8

G-4

8-0

9-6

11*2

12-8

14*4

17

1-7

3-4

5-1

G-8

8-5

10-2

11-9

13'6

15-3

18

1-8

3'6

6-4

7-2

9-0

10-8

12-6

14-4

16'2

19

1-9

3-8

5-7

7'G

9-5

11-4

13-3

16-2

17-1

20

2-0

4-0

6-0

8-0

10-0

12-0

14-0

16-0

18-0

21

2-1

4-2

6-3

8-4

10-5

12-6

14-7

16-8

18-9

22

2-2

4.4

6-6

8-8

11-0

13*2

15-4

17-6

19-8

23

2-3

4-6

G-9

9*2

11*5

13-8

1G-JL

18'4

20-7

24

2-4

4-8

7'2

9-6

12-0

14-4

16'8

19-2

21'6

25

2-5

5-0

7-5

10-0

12-5

15-0

17-6

20-0

22-5

26

2-6

5-2

7-8

10-4

13-0

15-6

18-2

20*8

23'4

27

2-7

5-4

8'1

10-8

135

16-2

18-9

21'6

243

28

2-8

5-6

8-4

11-2

14-0

16-8

19*6

22-4

25-2

29

2-9

5-8

8-7

11-6

14-5

17-4

20'3

23*2

26-1

30

3-0

6-0

9-0

12.0

15-0

18-0

21-0

24'0

27-0

31

3'1

6-2

9-3

12'4

15-5

18'6

21*7

24'8

27*9

32

3-2

6-4

9'G

12*8

16-0

19*2

22'4

25' G

28'8

33

3-3

G-6

9-9

13'2

1G-5

19-8

23-1

26-4

29*7

34

3-4

G-8

10-2

13'G

17-0

20-4

23-8

27*2

30-6

35

3'5

7'0

10-5

14-0

17-5

21-0

24-5

28-0

31-6

36

3'G

7-2

10-8

14-4

18-0

21-6

25-2

28-8

32'4

37

3-7

7-4

11-1

14-8

18-5

22-2

25-9

29-6

33-3

38

3'8

7-G

11-4

15-2

19-0

22-8

26-6

30-4

34-2

39

3'9

7-8

11-7

15-6

19-5

23-4

27'3

31-2

35-1

40

4-0

8-0

12-0

1G-0

20-0

24-0

28-0

32-0

36-0

41

4-1

8-2

12-3

1G-4

20-5

24-6

28-7

32-8

36'9

42

4-2

8-4

12-6

1G-8

21-0

25-2

29-4

33'6

37-8

43

4-3

8-6

12-9

17-2

21'5

25-8

30-1

34-4

38-7

44

4.4

8-8

13*2

17'6

22-0

26-4

308

35*2

39*6

45

4'5

9-0

13'5

18'0

22-5

27'0

31-5

36-0

40-5

46

4*6

9-2

13-8

18'4

23-0

27'6

32-2

36-8

41-4

47

4-7

9-4

14-1

18*8

23-5

28'2

32-9

37-6

42-3

48

4'8

9-6

14'4

192

24-0

28'8

33-6

38-4

43'2

49

4'9

9-8

14*7

19-6

24-5

29-4

34-3

9'2

44-1

50

5-0

10-0

15*0

20-0

25-0

30-0

35-0

*0'0

46-0

51

5-1

10-2

15-3

20-4

25-5

30-6

35-7

40*8

45-9

52

5-2

10-4

15*6

20-8

26-0

31-2

36'4

41-6

46-8

53

5*3

10-6

15'9

21-2

26-5

31-8

37*1

42'4

47'7

54

5-4

10'8

1G2

21-6

27-0

32*4

37-8

43-2

48*6

f

PROPORTIONAL PARTS continued.

X'l

X*2

x-3

x-4

x-5

x-6

x-7

x'8

x*9

55

5-5

11-0

1G-5

22-0

27-5

33-0

38-5

44-0

49-5

56

5-6

11-2

16-8

22-4

28-0

33-6

39-2

44-8

50-4

57

5-7

11-4

17-1

22-8

28*5

34-2

39'9

45'G

51-3

58

5-8

11-6

17-4

23-2

29-0

34-8

40'G

46-4

52-2

59

5-9

11-8

17-7

23-6

29-5

35-4

41-3

47-2

53-1

60

6-0

12-0

18'0

24-0

30-0

36-0

42-0

48-0

54-0

61

6-1

12-2

18-3

24-4

30-5

3G-G

42-7

48'8

54-9

62

6*2

12-4

18-6

24-8

31-0

37-2

43-4

49-6

55-8

63

6-3

12-6

18-9

25-2

31-5

37'8

44-1

50-4

5(5-7

64

6-4

12-8

19-2

25-6

82-0

38-4

44-8

51-2

57-G

65

6-5

13-0

19-5

2G-0

32-5

39-0

45-5

52-0

58-5

66

6-6

13-2

19-8

2G-4

33*0

39-6

4G-2

52-8

59-4

67

6-7

13-4

20-1

26-8

33-5

40-2

46-9

53-6

60-3

68

6-8

13-G

20-4

27'2

34-0

40-8

47-6

54-4

61-2

69

6-9

13-8

20-7

27-G

34-5

41-4

48*3

65'2

G2-1

70

7-0

14-0

21-0

28-0

35-0

42-0

49-0

56-0

G3-0

71

7-1

14-2

21-3

28-4

35-5

42-G

49-7

56-8

63-9

72

7-2

14-4

21-G

28-8

36-0

43-2

50-4

57-G

G4-8

73

7-3

14-6

21-0

29-2

3G-5

43-8

61-1

68-4

G5-7

74

7'4

14-8

22-2

29'G

37-0

44-4

51-8

59-2

GG-6

75

7-5

15-0

22-5

30-0

37-5

45-0

52-5

GO'O

67-5

76

7-6

15'2

22-8

30-4

38-0

45-6

5:5-2

60-8

68-4

77

7-7

15-4

23-1

30-8

38-5

46-2

53-9

61-6

G9-3

78

7-8

15-6

23-4

31-2

39-0

46-8

54-6

f.2-4

70-2

79

7-9

15-8

23-7

31-6

39 '5

47-4

55-3

G3-2

71*1

80

8-0

16-0

24-0

320

40-0

48-0

5G-0

64-0

72-0

81

8-1

16-2

24-3

32-4

40-5

48-G

56-7

C4-8

72-9

82

8-2

16*4

24-G

32-8

41-0

49-2

57-4

G5-6

73-8

83

8-3

16-6

24-9

33-2

41-5

49-8

58-1

GG-4

74-7

84

8-4

16-8

25-2

33-G

42-0

60-4

58-8

G7-2

76-6

85

8-5

17'0

25-5

34-0

42-5

51-0

69-5

G8'0

7G-5

86

8-6

17-2

25-8

34-4

43-0

51-6

60-2

68-8

77-4

87

8-7

17-4

26-1

34-8

43-5

52-2

60-9

G9-G

78-3

88

8-8

17-G

26-4

35-2

44-0

62-8

61-6

70-4

79-2

89

8-9

17-8

26-7

35'G

44-5

53-4

62'3

71-2

80-1

90

9-0

18-0

27-0

36-0

45-0

54'

G3'0

72-0

81-0

91

9-1

18-2

27-3

36'4

45-5

54-6

63-7

72-8

81-9

92

9-2

18-4

27-6

3G-8

46-0

55*2

64-4

73-6

82-8

93

9-3

18-6

27-9

37-2

46-5

55-8

65-1

74-4

83-7

94

9-4

18-8

28-2

37-6

47-0

5G-4

65-8

75-2

84-G

95

9-5

19-0

28-5

38-0

47*5

57-0

66-5

76-0

85-5

96

9*6

19*2

28-8

38-4

48-0

57-6

67-2

76-8

86-4

97

9-7

19'4

29-1

38-8

48-5

58-2

67-9

77-6

87-3

98

9-8

19-6

29-4

39-2

49-0

58-8

68-G

78-4

88-2

99

9'9

19-8

29-7

39-6

49-5

59-4

60-3

79-2

89-1

9 o

XXXIX. NUMBERS USED IN CALCULATIONS.

v = Ratio of Circumference of a Circle to Diameter.
e - Base of Natural Logarithms.

NUMBKHS.

TT = 3-14159 20536

2r= 0-28318 63072

47T = 12-50037 06144

\* = 1-57079 03208

|ir= 1-04719 75512

}* = 0-78539 81034

ITT^ 0-52359 87756

1^= 0-01745 32925

isio^- 0-00029 08882

(mfooo" O'OOOOO 48481

1= 4-18879 02048

l4-ir = 0-31830 98802

180 4- = 57-29577 95131

6-18000 4- TT = 200204-80625

1-4-4* = 0-07957 74715

1 -*- 67T = 0-05305 16477

*= 9-86960 44011

1 4- 7T 2 = 0-10132 11836

V*= 1-77245 38509

1 4- V = 56418 95835

y= 1-40459 18876

i!/7r 2 = 214502 93971

5/(64-)= 1-24070 09818

iy(C7T 2 )= 3-89777 70897

i = 4-83597 58020

e = 2-71828
Log 10 e= '43429
Log10 = 2-30258

= 1-0010005
: 1-0020020
= 1-0030045
= 1-0040080
: 1-0050125
: 1-0060180
: 1-0070245
= 1-0080320
: 1-0090405

LOGARITHMS.

0-49714 98727

0-79817 98684

1-09920 98640

0-19611 98770

0-02002 86180

1-89508 98814

1-71899 86223

2-24187 73676

4-46372 61172

G-08557 48068

0-02208 80093

1-50285 01273

1-75812 20324

5-31442 51332

2-90079 013GO

2-72469 88769

0-99429
1 00570
0-24857
1-75142
0-16571
0-33143
0-09366
0-59081
0-68448

97454
02516
49363
50637
66242
32485
71259
69986
41244

c .oos
e .oo9

18285

44819

50930

f .001 - -9990005
e -.oo2 = -9980020
e--w3 ~ -9970045
e ~.oo* = -9960080
e-(X)5 = -9950125
e- 000 = -9940180
e-w - -9930245
6 -.oos = -9920320
c-009 . -9910405
e -.ooor = l _

e = 2 3141 x 10

** = 5-3549 x 10 1
3 = 1-2392 x 10 4
e 4 * = 2-8675 x 10 6
e&r = 6-6356 X 10 8

V

e 2 ^ 4-8105
2 = 1-1132 x 10 1

6ir

2 = 2-5760 x 10 9
e2 = 5-9610 xlO<

9ir

2 = 1-3794 x 10"
e* = 21933

Sir

* = 1-0551 x 10

for

e 4 =5-0754x10
c v = 2-4415 x 10*

9>r

c 4 = 1-1745 xlO 3

llir

4 = 5-6498 x 10 3
c 4 " = 2-7178 x 10*
e 4 = 1-3074 x 10*
4 = 6-2893 x 10*

19rr

e 4 = 3-0205 x 10 6

- = 4-3214 x 10" 1
6-2, - 1-8674 x 10-
-fc = 8-0699 x 10'
-* = 3-4873 x 10~
e-5 = 1-5070 X 10f

e" 2 = 2-0788 x 10~>

Sir

" "2" =8-9833 xlO- 1

6ir

e" 2 = 3-8820 x 10" 4

_7ir

e" 2 = 1-6776 x 10~*
- 9ir
" 2 = 7-2495 x 10-'

tr

6"" 4 "= 4-5594 xlO- 1

Sir

e 4 = 9-4780 x 10- 8

5n-

e 4 " = 1-9703 x 10-*

- 7tr
i 4 = 4-0958 x 10-*

9?

c" * = 8-5144 x 10' 4

HIT
""*" =

= 1-7700 x 10~ 4

13*

T 4 = 3-6794 x 10"

15JT

e" 4 = 7-6487 X lO" 1

ITjr

" 4 = 1-5900 X 10^

I9rr

t 4 = 3-3053 x 10- T

EULBR'S NUMBERS B..

E 3 = 6-1 x 10
E, = 1-385 x 10*
E c = 5-0521 x 10 4

E 6 = 2-702765 x 10 fl
E 7 = 1-99360981 x 10 8
E 8 = 1-9391512145 x 10*
E 9 = 2-404879675441 x 1C 11

EULER'S CONSTANT

= 57721 56649

92

VALUES OP 8 W , s n , T., V w .

8, B 1-* -f 2- + 3"* -f . . , T n = 1- + 3~" + 5- 4- . . .
s= l - 2-" + 3"* - . . , V w = 1-* - 3"* 4- 5-* + . . .

n

s.

>

T.

I
2
3
4
5

00

1-64493 40668
1-20205 69032
1-08232 32337
1-03692 77551

69314 71806
82246 70334
90154 26774
94703 28295
97211 97704

00

1-23370 05501
1-05179 97903
1-01467 80316
1-00452 37628

6
7
8
9
10

1-01734 30620
1-00834 92774
1-00407 73562
1-00200 83928
1-00099 45751

98555 10913
-99259 38199
99623 30019
99809 42975
99903 95076

1-00144 70766
1-00047 15487
1-00016 51790
1-00005 13452
1-00001 70414

11

12
13
14
15

1-00049 41886
1-00024 60866
100012 27133
1-00006 12481
1-00003 05882

99951 71435
99975 76851
99987 85428
99993 91703
99996 95512

1-00000 56661
1-00000 18858
1-00000 06281
1-00000 02092
1-00000 00697

16
17

id

19
20

1-00001 52823
1-00000 76372
1-00000 38173
1-00000 19082
1-00000 09540

99998 47642
99999 23783
99999 61879
99999 80935
-99999 90466

1-00000 00232
1-00000 00077
1-00000 00026
1-00000 00009
1-00000 00003

ENGLISH AND METRIC EQUIVALENTS.

1 Metre (0 0.)
1 Square metre
1 Cubic decimetre
1 Litre
1 Kilogram
1 Gramme

lYard

1 Cubic inch

1 Gallon

1 Pound (avoir.)

= 39*370113 inches (62 F.)
= 10-7639 square feet
= 61*024 cubic inches
= 1*7598 pints
- 2'2046223 pounds (avoir.)
= 15-4323564 grains

= 0'914399 metres

= 16*387 cubic centimetres

= 4-5459631 litres

= 0-45359243 kilograms

PRINTED BY WILLIAM CLOWES AND SONS, LIMITED, LONDON AND BECCLES.

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